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THE 


AND 

ASTRONOMICAL  CALCUljATOR, 

Containing  the  Distances)  Diameters,  Periodical  and  Diurnal  Revolu- 
tions of  all  the  Planets  in  the  Solar  System,  with  the  Diameters  of  their 
Satellites)  their  distances  from,  and  the  periods  of  their  Revolutions 
around  their  respective  Primaries  $  together  with  the  method  of  calcula- 
ting those  Distances^  Diameters  and  Revolutions*  Also  the  method  of 
calculating  Solar  and  Lunar  Eclipses ;  being  a  compilation  from  various 
celebrated  authors,  with  Notes,  Examples  and  Interrogations  )  prepared 
for  the  use  of  Schools  and  Academies* 

BY 
TOBIAS   OSTRANI>ER, 

TJSMCHJEM  OF  MATHEMATICS, 

AND 

AUTHOR  OF  THE  ELEMENTS  OF  NUMBERS,  EASY  INSTRUCTOR, 
MATHEMATICAL  EXPOSITOR,  &C. 

"Consult  with  reason,  reason  will  reply, 

"Each  lucid  point,  which  glows  in  yonder  sky, 

"Informs  a  system  in  the  boundless  space, 

« And  fills  with  glory  its  appointed  place ; 

"With  beams  unborrow'd  brightens  other  skies, 

"And  worlds,  to  thee  unknown,  with  heat  and  life  supplies." 


PRINTED  AT    THE    OFFICE   OF    THE    WESTERN 

1832, 


Nerthtrn  Dittrlet  of  New- York,  TO  WIT: — 

BE  it  remembered,  that  on  the  sixteenth  day  of  July,  Anno  Domini, 
1832,  TOBIAS  OSTRANDER  of  the  said  district,  hath  deposited  in 
this  office  the  title  of  a  book,  the  title  of  which  is  in  the  words  following, 
to  wit : 

The  Planetarium  and  Astronomical  Calculator,  containing  the  Distan- 
ces, Diameters,  Periodical  and  Diurnal  Revolutions  of  all  the  Planets  in 
the  Solar  system,  with  the  Diameters  of  their  Satellites,  their  distances 
from,  and  the  periods  of  their  Revolutions  around  their  respective  Prima- 
ries ;  together  with  the  method  of  calculating  those  Distances,  Diameters 
and  Revolutions  ;  and  the  method  of  calculating  Solar  and  Lunar  Eclipses  ; 
being  a  compilation  from  various  celebrated  authors,  with  Notes,  Exam- 
ples and  Interrogations ;  prepared  for  the  use  of  Schools  and  Academies. 
By  TOAIAS  OSTRANDER,  Teacher  of  Mathematics,  and  author  of  the 
Elements  of  Numbers,  Easy  Instructor,  Mathematical  Expositor,  &c. 

"Consult  with  reason,  reason  will  reply, 
"  Each  lucid  point  which  glows  in  yonder   sky, 
;    "Informs  a  system  in  the  boundless  space, 
"  And  fills  with  glory  its  appointed  place ; 
"With  beams  unborrow'd  brightens  other  skies, 
"And  worlds,  to  thee  unknown,  with  heat  and  life  supplies." 

The  right  whereof  he  claims  as  Compiler  and  Proprietor,  in  confer- 
ity  with  an  act  of  Congress,  entitled  an  Act  to  amend  the  several  Acts 
respecting  Copy  Rights. 

•      RUTGER  B.  MILLER, 
Cltrk  of  the  Northern  District  of  New-York. 


I^ 


3* 


PREFACE. 

IN  presenting  the  following  pages  to  the  public,  I  will 
briefly  remark,  that  the  people  generally  are  grossly 
ignorant  in  the  important  and  engaging  science  of  As- 
tronomy. Scarcely  one  in  a  county  is  found  capable 
of  calculating  with  exactness,  and  accuracy  the  precise 
time  of  an  eclipse,  or  conjunction  and  opposition  of  the 
Sun  and  Moon.  Is  it  for  lack  of  abilities  ?  No. — 
There  are  no  people  on  the  surface  of  this  terraqueous 
globe,  who  possess  better  natural  faculties  of  acquiring 
knowledge  of  any  description,  than  those  who  inhabit 
the  United  States  of  America.  In  this  land  of  liberty, 
much  has  been  done,  and  much  still  remains  to  be 
done,  for  the  benefit  of  the  rising  generation.  Schools, 
Academies  and  Colleges  have  been  erected,  for  the 
purpose  of  facilitating,  and  extending  information  and 
instruction  among  the  youth  of  this  delightful  section. 
Gentlemen  possessing  the  most  profound  abilities  and 
acquirements,  have  engaged  in  the  truly  laudable  em- 
ployment of  disseminating  a  knowledge  of  all  the  scien- 
ces ;  both  of  useful  and  ornamental  description.  Still  this 
branch  of  the  Mathematical  science,  called  Astronomy, 
has  been  almost  totally  neglected,  especially  among  the 
common  people.  From  what  source  has  this  origina- 
ted 1  I  answer,  from  a  scarcity  of  books,  well  calcu- 


M288838 


lated  to  give  the  necessary  instruction.  Though  there 
are  many  productions  possessing  merit,  and  are  of 
importance  to  the  rising  generation,  yet  they  are  defi- 
cient in  the  tables  necessary  for  the  calculation,  and 
protraction  of  eclipses.  The  works  of  Ferguson,  En- 
field  and  others,  from  which  this  is  principally  com- 
piled, contain  all  that  is  necessary  ;  but  the  expense 
renders  them  beyond  the  means  of  many,  who  perhaps 
posses  the  best  abilities  in  our  land.  Extensive  vol- 
umes are  not  well  calculated  for  the  use  of  Schools  ; 
for  a  Student  is  under  the  necessity  of  reading  so  much 
unessential,  and  uninteresting  matter,  that  the  essence 
is  lost,  in  the  multiplicity  of  words  ;  and  for  these  rea- 
sons, many  of  the  teachers  have  neglected  this  useful, 
and  important  branch  of  the  Mathematical  science.  I 
have  long  impatiently  beheld  the  evil,  without  an  op- 
portunity of  providing  a  remedy,  until  the  present 
period. 

I  now  present  to  this  enlightened  community,  a 
volume  within  the  means  of  almost  every  person ;  con- 
taining all  the  essential  parts  of  Astronomy,  adapted 
to  the  use  of  Schools  and  Academies  ;  made  so  plain 
and  easy  to  be  understood,  that  a  lad  of  twelve  years 
of  age,  whose  knowledge  of  Arithmetic  extends  to  the 
single  rule  of  proportion,  can,  in  the  short  space  of  one 
or  two  weeks,  be  taught  to  calculate  an  eclipse ;  and 
many  possessing  riper  years,  from  the  precepts  and 
examples  given  in  the  work,  will  be  found  capable  of 
accomplishing  it,  without  the  aid  of  any  other  teacher, 


The  tables,  (with  the  exception  of  two,)  I  have 
wholly  calculated,  and  then  duly  compared  them  with 
those  of  Ferguson.  Great  care  has  also  been  taken, 
to  present  the  work  to  the  public,  free  from  errors. 

Should  the  following  pages  meet  the  approbation  of 
a  generous  and  enlightened  community,  and  be  the 
means  of  extending  the  knowledge  of  this  important 
branch  of  Education ;  not  only  to  the  rising  generation, 
but  to  those  of  maturer  years,  the  Compiler,  whose 
best  abilities  have  hitherto  been  employed  in  endeav- 
oring to  meliorate  the  condition  of  man,  by  improving 
the  mind  and  enlightening  the  understanding,  will  have 
the  sublime  satisfaction,  of  removing  some  of  the 
shackles  of  ignorance,  and  building  up  a  fund  of  useful 
and  interesting  knowledge  upon  its  ruins. 

THE  COMPILER. 


SECTION  FIRST. 

OF  ASTRONOMY  IN  GENERAL. 

OF  all  the  sciences  cultivated  by  mankind,  Astron- 
omy is  acknowledged  to  be,  and  undoubtedly  is,  the 
most  sublime,  the  most  interesting,  and  th^most  use- 
ful. By  the  knowledge  derived  from  this  science,  not 
only  the  magnitude  of  the  earth  is  discovered,  the  sit- 
uation and  extent  of  the  Countries  and  Kingdoms  as- 
certained, trade  and  commerce  carried  on  to  the  re- 
motest parts  of  the  world,  and  the  various  products  of 
several  countries  distributed,  for  the  health,  comfort, 
and  conveniency  of  its  inhabitants ;  but  our  very  facul- 
ties are  enlarged,  with  the  grandeur  of  the  ideas  it  con- 
veys, our  minds  exalted  above  the  low  contracted  pre- 
judices of  the  vulgar,  and  our  understandings  clearly 
convinced,  and  affected  with  the  conviction,  of  the  ex- 
istence, wisdom,  power,  goodness,  immutability,  and 


8  Of  Jlstronomy  in  General.  Sec.  1. 

superintendency  of  the  Supreme  Being.  So  that 
without  any  hyperbole,  every  man  acquainted  with 
this  science,  must  exclaim  with  the  immortal  Dr. 
Young :  "  An  undevout  Astronomef  is  mad."  From 
this  branch  of  Mathematical  knowledge,  we  also  learn 
by  what  means,  or  laws,  the  Almighty  Power  and 
Wisdom  of  the  Supreme  Architect  of  the  Uuiverse, 
are  administered  in  continuing  the  wonderful  harmony, 
order  and  connexion,  observable  throughout  the  plan- 
etary system ;  and  are  led  by  very  powerful  arguments, 
to  form  this  pleasing  and  cheering  sentiment,  that 
minds  capable  of  such  deep  researches,  not  only  derive 
their  origin  from  that  Adorable  Being,  but  are  also 
incited  to  aspire  after  a  more  perfect  knowledge  of 
his  nature,  and  a  more  strict  conformity  to  his  will. 

By  Astronomy  we  discover,  that  the  earth  is  at  so 
great  a  distance  from  the  sun,  that  if  seen  from  thence, 
it  would  appear  no  larger  than  a  point ;  although  its 
diameter  is  known  to  be  nearly  8,000  miles :  yet  that 
distance  is  so  small,  compared  with  the  earth's  distance 
from  the  fixed  stars,  that  if  the  orbit,  in  which  the 
earth  moves  round  the  sun,  were  solid,  and  seen  from 
the  nearest  star,  it  would  likewise  appear  no  larger 
than  a  point;  although  it  is  at  least  190  millions  of 
miles  in  diameter ;  for  the  earth  in  going  round  the 
sun,  is  190  millions  of  miles  nearer  to  some  of  the 
stars,  at  one  time  of  the  year  than  at  another ;  and  yet 
their  apparent  magnitudes,  situations,  and  distances 
still  remain  the  same  ;and  a  telescope  which  magnifies 
above  200  times,  does  not  sensibly  magnify  them; 


Sec.  I  Of  Astronomy  in  General.  9 

which  proves  them  to  be  at  least,  one  hundred  thou- 
sand times  further  from  us,  than  we  are  from  the  sun. 

It  is  not  to  be  imagined,  that  all  the  stars  are  placed 
in  one  concave  surface,  so  as  to  be  equally  distant  from 
us;  but  that  they  are  placed  at  immense  distances 
from  one  another,  through  unlimited  space,  so  that 
there  may  be  as  great  a  distance  between  any  two 
neighboring  stars,  as  between  the  sun  from  which  we 
receive  our  light,  and  those  which  are  nearest  to  him. 
Therefore,  an  observer  who  is  nearest  any  fixed  star, 
will  look  upon  it  alone  as  a  real  sun  ;  and  consider  the 
rest  as  so  many  shining  points,  placed  at  equal  distances 
from  him  in  the  firmament, 

By  the  help  of  telescopes,  we  discover  thousands  of 
stars  which  are  entirely  invisible,  without  the  aid  of 
such  instruments,  and  the  better  our  glasses  are,  the 
more  become  visible.  We  therefore  can  set  no  limits 
to  their  numbers,  or  to  their  immeasurable  distances. 
The  celebrated  Huygens  carried  his  thoughts  so  far, 
as  to  believe  it  not  impossible,  that  there  may  be  stars 
at  such  inconceivable  distances,  that  their  light  has  not 
yet  reached  the  earth  since  their  creation ;  although 
the  velocity  of  light,  be  a  million  of  times  greater 
than  the  velocity  of  a  cannon  ball  at  its  first  discharge ; 
and  as  Mr.  Addison  justly  observes,  "  This  thought  is 
far  from  being  extravagant,  when  we  consider  that  the 
Universe  is  the  work  of  Infinite  Power,  prompted  by 
Infinite  Goodness,  and  having  an  Infinite  space  to  exert 
itself  in ;  therefore  our  finite  imaginations  can  set  no 
bounds  to  it." 

A 


10  Of  Astronomy  in  General  Sec.  1. 

The  Sun  appears  very  bright  and  large,  in  compar- 
ison of  the  fixed  stars ;  because  we  constantly  keep 
near  the  Sun,  in  comparison  to  our  immense  distance 
from  them.  For  a  spectator  placed  as  near  to  any 
star,  as  we  are  to  the  Sun,  would  see  that  star  to  be  a 
body  as  large  and  bright  as  the  Sun  appears  to  us  : 
and  a  spectator  as  far  distant  from  the  Sun,  as  we  are 
from  the  stars,  would  see  the  Sun  as  small  as  we  see 
a  star,  divested  of  all  its  circumvolving  planets,  and 
would  reckon  it  one  of  the  stars,  in  numbering  them. 

The  stars  being  at  such  immense  distances  from  the 
Sun,  cannot  possibly  receive  from  him  so  strong  alight 
as  they  appear  to  have,  nor  any  brightness  sufficient 
to  make  them  visible  to  ua ;  for  the  Sun's  rays  must 
be  so  scattered  before  they  reach  such  remote  objects, 
that  they  can  never  be  transmitted  back  to  our  eyes  ; 
so  as  to  render  these  objects  visible  by  reflection. — 
Therefore  the  stars,  like  the  Sun,  shine  with  ther  own 
native  and  unborrowed  lustre ;  and  since  each  particular 
one,  as  well  as  the  Sun,  is  confined  to  a  particular  por- 
tion of  space,  it  is  evident  that  the  stars  are  of  the  same 
nature  with  the  Sun;  formed  of  similar  materials,  and 
are  placed  near  the  centres  of  as  many  magnificent 
systems ;  have  a  retinue  of  worlds  inhabited  by  intelli- 
gent beings,  revolving  round  them  as  their  common 
centres ;  receive  the  distribution  of  their  rays,  and  are 
illuminated  by  their  beams;  all  of  which,  are  losttous, 
in  immeasurable  wilds  of  ether. 

It  is  not  probable  that  the  Almighty,  who  always 
acts  with  Infinite  Wisdom,  and  does  nothing  in  vain, 


Sec*  1  Of  Astronomy  in  General.  1 1 

should  create  so  many  glorious  Suns,  fit  for  so  many 
important  purposes,  and  place  them  at  such  distances 
from  each  other  without  proper  objects  near  enough 
to  be  benefit-ted  by  their  influences.  Whoever  ima- 
gines that  they  were  created  only  to  give  a  faint  glim- 
mering light  to  the  inhabitants  of  this  globe,  must  have 
a  very  superficial  knowledge  of  Astronomy,  and  a  mean 
opinion  of  the  Divine  Wisdom  ;  since  by  an  infinitely 
less  exertion  of  creating  power,  the  Deity  could  have 
given  our  earth  much  more  light  by  one  single  addi- 
tional Moon. 

Instead  of  cur  Sun,  and  our  world  only  in  the  Uni- 
verse, (as  the  unskillful  in  Astronomy  may  imagine  ;) 
that  science  discovers  to  us,  such  an  inconceivable 
number  of  Suns,  Systems  and  Worlds,  dispersed 
through  boundless  space,  that  if  our  Sun,  with  all  the 
planets,  Moons  and  Comets,  belonging  to  the  whole  So- 
lar System,  were  at  once  annihilated,  they  would  no 
more  be  missed  by  an  eye  that  could  take  in  the  whole 
compass  of  Creation,  than  a  grain  of  sand  from  the  Sea 
shore ;  the  space  they  possess,  being  comparatively  so 
small,  that  their  loss  would  scarcely  make  a  sensible 
blank  in  the  Universe.  Although  Herschel,  the  out- 
ermost of  our  planets,  revolves  about  the  Sun,  in  an 
orbit  of  three  thousand,  six  hundred  millions  of  miles 
in  diameter,  and  some  of  our  Comets,  make  excursions 
more  than  ten  thousand  millions  of  miles  beyond  his  or- 
bit, and  yet  at  that  amazing  distance,  they  are  incom- 
parably nearer  the  Sun,  than  to  any  of  the  fixed  stars, 
as  is  evident,  from  their  keeping  clear  of  the  attractive 


12  Of  Astronomy  in  General. 

power  of  all  the  stars,  and  returning  periodically  by 
virtue  of  the  Sun's  attraction. 

From  what  we  know  of  our  own  System,  it  may  be 
reasonably  concluded,  that  all  the  rest  are  with  equal 
wisdom  contrived,  situated  and  provided  with  accom- 
modations for  the  existence  of  intelligent  inhabitants. 
Let  us  therefore  take  a  survey  of  the  System  to  which 
we  belong,  the  only  one  accessible  to  us,  and  from 
thence  we  shall  be  better  able  to  judge  of  the  nature 
and  end  of  other  systems  of  the  Universe.  Altho'  there 
are  almost  an  infinite  variety  in  the  parts  of  Creation, 
which  we  have  opportunities  of  examining  ;  yet  there 
is  a  general  analogy  running  ihrough,  and  connecting 
all  the  parts  into  one  scheme ,  one  design  of  dissemmi- 
nating  comfort  and  happiness  to  the  whole  Creation. — 
To  an  attentive  observer,  it  will  appear  highly  proba- 
able,  that  the  planets  of  our  System,  together  with  their 
attendants  called  Satellites  or  Moons,  are  much  of  the 
same  nature  with  our  earth,  and  destined  for  similar 
purposes  ;  for  they  are  all  solid  opaque  globes,  capable 
of  supporting  animals  and  vegetables  ;  some  are  larg- 
er, some  less,  and  one  nearly  the  size  oi  our  earth. — 
They  all  circulate  round  the  Sun,  as  the  earth  does, 
in  a  shorter,  or  longer  time,  according  to  their  res- 
pective distances  from  him,  and  have,  where  it  would 
not  be  inconvenient,)  regular  returns  of  Summer  and 
Winter,  Spring  and  Autumn.  They  have  warmer 
and  colder  climates,  as  the  various  productions  of  our 
earth  require,  and  of  such  as  afford  a  possibility  of  dis- 
covering it,  we  observe  a  regular  motion  round  their 


Sec.  1  Of  Astronomy  in  General.  13 

axes,  like  that  of  our  earth,  causing  an  alternate  return 
of  day  and  night,  which  is  necessary  for  labour,  rest, 
and  vegetation,  and  that  all  parts  of  their  surfaces  may 
be  exposed  to  the  rays  of  the  Sun. 

Such  of  the  planets  as  are  farthest  from  the  Sun,  and 
therefore  enjoy  least  of  his  light,  have  that  deficiency 
made  up  by  several  Moons,  which  constantly  accom- 
pany and  revolve  about  them,  as  our  Moon  revolves 
around  the  earth.  The  planet  Saturn  has  over  and 
above  a  broad  ring,  encompassing  it,  which  no  where 
touches  his  body;  which  like  a  broad  zone  in  the 
Heavens,  reflects  the  same  light  very  copiously  on 
that  planet  ;  remote  planets  have  the  Sun's  light  fainter 
by  day,  than  we,  they  have  an  addition  to  it,  morning 
and  evening,  by  one  or  more  of  their  Moons,  and  a 
greater  quantity  of  light  in  the  night  time. 

On  the  surface  of  the  Moon,  (because  it  is  nearer  to 
us  than  any  other  of  the  celestial  bodies,)  we  discover 
a  nearer  resemblance  of  our  earth,  for  by  the  assistance 
of  telescopes  we  observe  the  Moon  to  be  full  of  high 
mountains,  large  vallies,  and  deep  cavities.  These  sim- 
ilarities leave  us  no  room  to  doubt,  but  that  all  planets, 
Moons  and  Systems,  are  designed  to  be  commodious 
habitations  for  creatures  endowed  with  capacities  of 
knowing,  and  adoring  their  beneficent  Creator. 

Since  the  fixed  stars  are  prodigious  spheres  shining 
by  their  own  native  light  like  our  Sun,  at  inconceiv- 
able distances  from  each  other,  as  well  as  from  us,  it 
is  reasonable  to  conclude  that  they  are  made  for  simi- 
lar purposes,  each  to  bestow  light,  heat  and  vegetation 


14  Of  Astronomy  in  General.  Sec.  1 

on  a  certain  number  of  inhabited  planets,  kept  by  grav- 
itation within  the  sphere  of  its  activity. 

When  we  therefore  contemplate  on  those  ample  and 
amazing  structures,  erected  in  endless  magnificence 
over  all  the  ethereal  plains,  when  \ve  look  up- 
on them  as  so  many  repositories  of  light,  or  fruitful 
abodes  of  life,  when  we  consider  that  in  all  probability 
there  are  orbs  vastly  more  remote  than  those  which 
appear  to  our  unaided  sight,  orbs  whose  effulgence, 
though  travelling  ever  since  the  Creation,  has  not  yet 
arrived  upon  our  coast — 

What  an  august,  what  an  amazing  conception  does 
this  give  of  the  works  of  the  Omnipotent  Creator  ; 
who  made  use  of  no  preparatory  measures,  or  long 
circuit  of  means.  He  spake,  and  ten  thousand  times 
ten  thousand  Suns,  multiplied  without  end,  hanging 
pendulous  in  the  great  vault  of  Heaven,  at  immense 
distances  from  each  other,  attended  by  ten  thousand 
times  ten  thousand  worlds,  all  in  rapid  motion,  yet  calm, 
regular,  and  harmonious,  invariably  keeping  the  paths 
prescribed,  rolled  from  his  creating  hand. 

But  when  we  contemplate  on  the  power,  wisdom, 
goodness,  and  magnificence  of  the  Great  Creator  ;  let 
us  use  the  language  of  the  immortal  Dr.  Young,  in  his 
appeal  to  the  starry  Heavens  : — 


-"  Say  proud  arch, 


Built  with  Divine  ambition,  in  disdain 
Oflimit  built ;  built  in  the  taste  of  Heaven, 
Vast  concave,  ample  dome.    Wast  thou  designed 
A  meet  apartment  for  the  Deity  ? 
Not  so ;  that  thought  alone  thy  state  impairs, 
Thy  lofty  sinks,  and  shallows  thy  profound, 
And  straightens  thy  diffusive." 


Sec.  1  Interrogations  for  Section  First.  15 

INTERROGATIONS  FOR  SECTION  FIRST. 

What  is  ASTRONOMY  ? 

It  is  a  mixed  Mathematical  Science,  teaching  the 
knowledge  of  the  celestial  bodies,  their  magnitudes, 
motions,  distances,  periods,  eclipses  and  order. 

What  are  its  uses  1 

What  conviction  does  a  knowledge  of  this  branch  of 
science  give  to  the  understanding  1 

What  cheering  sentiment  is  formed  from  a  knowl- 
edge of  this  science  1 

What  is  the  diameter  of  the  earth  ? 

How  many  miles  is  the  diameter  of  the  earth's  orbit  ? 

How  is  it  known  that  the  stars  are  at  immense  dis- 
tances from  us  ? 

How  is  it  known  that  they  are  at  immense  distances 
from  each  other  1 

What  instruments  have  been  invented  to  aid  the 
sight  of  man  1 

Who  supposed  there  were  stars,  whose  light  had 
not  yet  reached  the  earth  since  their  first  creation],? 

Who  confirmed  the  idea  ? 

Why  cannot  the  same  rays  be  reflected  back  from 
the  stars  to  our  eyes  1 

With  what  light  do  the  stars  shine  1 

How  could  the  Deity  have  given  us  greater  light 
in  the  night  time,  than  by  the  wrhole  starry  host? 

How  is  it  known  that  the  Comets  belong  to  the  So- 
lar Svstem  7 


1 6  Interrogations  for  Section  First.  Sec.  I 

From  what  parity  of  reasoning,  is  it  believed  that 
the  stars  are  so  many  suns,  and  have  worlds  revolving 
about  them  ? 

How  are  those  planets  supplied  with  light,  which 
are  farthest  from  the  Sun  ? 


SECTION  SECOND. 
OF  THE  SOJL*£& 


THE  Solar  System  consists  of  the  Sun,  with  all  the 
Planets  and  Comets  that  move  around  him  as  their 
centre.  Those  which  are  near  the  Sun,  not  only  fin- 
ish their  circuits  sooner,  but  likewise  move  with  great- 
er rapidity  in  their  respective  orbits,  than  those  which 
are  more  remote.  Their  motions  are  all  performed 
from  West  to  East,  in  Elliptical  orbits.  Their  names, 
distances,  magnitudes,  and  periodical  revolutions,  are 
as  follows  :  —  The  Sun  is  placed  near  the  common  cen- 
tre, or  rather  in  the  lower  focus  of  the  orbits  of  all  the 
planets  and  comets,  and  turns  round  on  his  axis  once 
in  25  days,  14  hours  and  8  minutes;  as  has  been  pro- 
ved, from  the  motion  of  the  spots,  seen  on  his  surface. 
His  diameter  is  computed  at  883,246  miles,  and  by  the 
various  attractions  of  the  convolving  planets,  he  is  agi- 
tated by  a  small  motion  round  the  centre  of  gravity  of 
the  system.  His  mean  apparent  diameter  as  seen 
from  the  earth,  is  32  minutes  and  one  second.  His  *o- 

B 


18  Of  the  Solar  System.  Sec.  2. 

lidity,  and  indeed  that  of  every  other  planet,  may  be 
found  by  multiplying  the  cube  of  their  diameters  by 
,5236.  All  the  planets  as  seen  by  a  spectator,  placed 
on  the  sun,  move  the  same  way,  and  according  to  the 
order  of  the  signs,  Aries,  Taurus,  Gemini,  &e.  which 
represent  the  great  ecliptic.  But,  to  a  spectator  placed 
on  any  one  of  the  planets,  the  others  sometimes  appear 
to  go  backward,  sometimes  forward,  and  at  others  sta- 
tionary ;  not  moving  in  proper  circles,  nor  elliptical  or- 
bits, but  in  looped  curves,  which  never  return  into  them- 
selves. 

The  Comets,  also  appear  to  come  from  all  parts,  and 
appear  to  move  in  various  directions.  These  proofs 
are  sufficient  to  establish  the  fact,  that  the  sun  is  placed 
near  the  centre,  and  that  all  the  other  planets  revolve 
around  him  :  are  irradiated  by  his  beams  :  receive  the 
distribution  of  his  rays,  and  are  dependant  for  the  enjoy- 
ment of  every  blessing  on  this  grand  dispensor  of  divine 
munificence. 

The  orbits  of  the  planets  are  not  in  the  same  plane 
with  the  ecliptic,*  but  crosses  it  in  two  points  directly 
opposite  to  each  other,  called  the  planet's  nodes,  f — 
That  from  which  the  planet  ascends  northward  above 
the  ecliptic,  is  called  the  ascending  node ;  and  the  other 
which  is  directly  opposite,  (and  consequently  6  signs 
asunder,)  is  called  the  descending  node. 

*  The  ecliptic  is  an  imaginary  great  circle  in  the  Heavens,  in  the  plane 
of  which  the  earth  performs  her  annual  revolutions  round  the  sun. 

t  The  node  is  the  intersection  of  the  orbit  of  any  planet  with  that  of  the 
••rtfc. 


Sec.  2  Of  the  Solar  System.  19 

It  was  discovered  on  the  first  of  January  1805,  that 
the  ascending  node  of  the  planet  Herschel  was  in  twelve 
degrees  and  fifty -three  minutes  of  the  sign  Gemini,  and 
advances  16  seconds  in  a  year.  Saturn  in  twenty-one 
degrees  and  59  minutes  of  Cancer,  and  advances  32 
seconds  in  a  year.  Jupiter  in  8  degrees  and  27  minutes 
of  Cancer,  and  advances  36  seconds  yearly.  Mars  in 
18  degrees,  and  four  minutes  of  Taurus,  and  advances 
28  seconds  yearly.  Venus  in  14  degrees  and  55  min- 
utes of  Gemini,  and  advances  36  seconds  yearly.  Mer- 
cury in  16  degrees  of  Taurus,  and  advances  43  seconds 
every  year.  In  these  observations,  the  earth's  orbit  is 
considered  the  standard,  and  the  orbits  of  all  the  ether 
planets  obliquely  to  it.  The  nearest  planet  to  the  sun 
is  Mercury.  The  great  brilliancy  of  light  emitted  by 
this  planet:  the  shortness  of  the  period  during  which 
observations  can  be  made  upon  his  disk ;  and  his  posi- 
tion among  the  vapors  of  the  horizon  when  he  is  obser- 
ved, have  hitherto  prevented  Astronomers,  from  ma- 
king interesting  discoveries  to  be  relied  on  with  cer- 
tainty respecting  this  planet.  This  planet,  when  view- 
ed at  different  times  with  a  good  telescope,  appears  in 
all  the  various  shapes  of  the  Moon,  which  is  a  plain 
proof  that  he  receives,  (like~  the  Moon,)  all  his  light 
from  the  Sun.  That  he  moves  round  the  Sun  in  an 
orbit,  within  the  orbit  of  the  earth,  is  also  plain  ;  be- 
cause he  is  never  seen  opposite  to  the  Sun,  nor  above 
56  times  the  Sun's  diameter  from  his  centre.  It  has 
been  said  by  Authors,  that  his  light  and  heat  from 
the  Sun  must  be  almost  seven  times  as  great  as  our's ; 


20  Of  the  Solar  System.  Sec.  2 

judging  from  his  nearness  to  it.  His  light  and  heat 
however,  depend  more  on  the  height  and  density  of 
his  atmosphere,  than  to  his  near  approach  to  that  lu- 
minary. 

His  distance  from  the  Sun  is  computed  at  73,000,000 
of  miles,  is  3,225  in  diameter,  and  performs  a  revolu- 
tion round  the  Sun,  in  87  days  23  hours  15  minutes  and 
28  seconds  :  his  apparent  diameter  as  seen  from  the 
earth,  is  ten  seconds.  His  orbit  is  inclined,7  degrees 
to  the  ecliptic  ;  and  that  node  from  which  he  ascends 
northward  above  it,  is  in  the  1 6th  degree  of  Taurus,  the 
opposite  in  the  16th  degree  of  Scorpio.  The  earth  is 
in  these  points  on  the  6th  of  November,  and  4th  of 
May;  when  he  comes  to  either  of  his  nodes  at  his  infe- 
rior conjunction  about  these  times,  he  will  appear  to 
pass  over  the  face  of  the  sun  like  a  dark  round  spot. — 
But  in  all  other  parts  of  his  orbit,  his  conjunctions  are 
invisible  ;  because  he  either  goes  above  or  below  the 
Sun.  On  the  5th  day  of  May,  at  6  hours  43  minutes 
22  seconds  in  the  morning,  in  the  year  1832,  in  the 
longitude  of  Washington,  he  was  in  conjunction  with 
the  Sun.  His  next  visible  conjunction  will  be  on  the 
7th  day  of  November  1835. 

Venus,  the  next  planet  in  order  is  68,000,000  of 
miles  from  the  Sun  by  computation,  and  by  moving  at 
the  rate  of  69,000  miles  every  hour  in  her  orbit,  she 
performs  her  revolution  round  the  Sun  in  224  days,  16 
hours  and  49  minutes  of  our  time  ;  in  which,  (though 
it  be  the  full  length  of  her  year,)  she  has  only  9  days 
and  a  quarter,  according  to  observations  made  by  Bi- 


Sec.  2  Of  the  Solar  System.  2 1 

anchini ;  so  that  her  every  day  and  night  together,  is  as 
long  as  348^  days  and  nights  with  us.  This  odd  quar- 
ter of  a  day  in  every  year,  makes  in  every  fourth  year 
a  leap  year  to  Venus,  as  the  like  does  to  the  earth 
which  we  inhabit.  Her  diameter  is  computed  at  7687 
miles,  and  performs  her  diurnal  revolutions  in  23  hours 
20  minutes,  and  54  seconds  ;  with  an  inclination  ol  her 
orbit  to  the  ecliptic,  of  3  degrees  23  minutes,  and  35 
seconds.  Her  orbit  includes  the  orbit  of  Mercury 
within  it,  for  at  her  greatest  elongation,  or  apparent 
distance  from  the  Sun,  she  is  about  96  times  his  diam- 
eter from  his  centre  ;  while  that  of  Mercury  is  not 
above  56. 

Her  orbit  is  included  within  the  orbit  of  the  earth, 
for  if  it  were  not,  she  would  be  as  often  seen  in  oppo- 
sition as  in  conjunction  with  the  Sun.  But  she  never 
departs  from  the  Sun  to  exceed  47  degrees,  and  that 
of  Mercury  28,  it  is  therefore  certain  that  the  orbit  of 
Mercury  is  within  the  orbit  of  Venus,  and  that  of  Ve- 
nus within  the  orbit  of  the  earth.  When  this  planet  is 
west  of  the  Sun,  she  rises  in  the  morning  before  him, 
and  hence  she  is  called  the  morning  star  ;  and  when 
she  sets  after  the  Sun,  she  is  called  the  evening  star  ; 
so  that  in  one  part  of  her  orbit  she  rides  foremost  in 
the  procession  of  night,  and  in  the  othe'r,  anticipates  the 
dawn  ;  being  each  in  its  turn  290  days.  The  axis  of 
Venus  is  inclined  75  degrees  to  the  axis  of  her  orbit, 
which  is  51  degrees  and  32  minutes  more  than  the 
axis  of  the  earth  is  inclined  to  the  axis  of  the  ecliptic  ; 
and  therefore  her  seasons  vary  much  more  than  our's. 


22  Of  the  Solar  System.  Sec.  2 

The  north  pole  of  her  axis,  inclines  towards  the  20th 
degree  of  Aquarius;  the  earth's  to  the  beginning  of 
Cancer.  Consequently  the  northern  parts  of  Venus 
have  Summer,  in  the  signs  where  those  of  the  earth 
have  Winter,  and  vice  versa.  The  orbit  of  Venus  is 
inclined  three  and  one  half  degrees  to  the  earth's,  and 
crosses  it  in  the'l  4th  degree  of  Gemini,  and  Sagittarius, 
and  therefore  when  the  earth  is  near  the  points  of  the 
ecliptic,  at  the  time  when  Venus  is  in  her  inferior  con- 
junction,* she  appears  like  a  spot  on  the  Sun,  and  it 
furnishes  a  true  method  of  calculating  the  distances  of 
all  the  planets  from  the  Sun. 

•It  will  not  be  uninteresting  to  those  who  peruse  this 
treatise,  to  be  put  in  the  possession  of  all  the  elements 
of  the  transits,  both  of  Mercury  and  Venus  over  the 
Sun's  disk  ;  from  this  period  to  the  end  of  the  present 
century,  I  therefore  insert  the  following  tables  : — 


TRANSIT  OF  MERCURY  OVER  THE  SUN'S  DISK. 
Transit  of  Mercury  May  4th  1832. 

D    H.    M.    8. 

Mean  time  of  conjunction,  May         -  4  23  51  22 

8   D.    M.    S. 

Geocentric  longitude  of  the  Sun  and  Mercury       -       1  14  56  45 

H    M.    8. 

Middle  apparent  time,  0   18     1 

Semi-duration  of  the  transit,  3  28    2 

Nearest  approach  to  centres,         -  8  16  North. 


*  Inferior  conjunction  is,  when  the  planet  is  between  the  earth  and  the 
Sun,  in  the  nearest  part  of  its  orbit. 


Sec.  2 


Of  the  Solar  System. 

November  7th  1835. 


Mean  time  of  conjunction, 

Geocentric  longitude  of  the  Sun  and  Mercury, 

Middle  apparent  time, 
Semi  duration  of  the  transit, 
Nearest  approach  of  centres, 

May  Sth  1845, 
Mean  time  of  Conjunction. 
Geocentric  longitude  of  the  Sun  and  Mercury, 

Middle  apparent  time, 
Semi-duration  of  the  transit, 
Nearest  approach  of  centres, 

November  9th  1848. 
Mean  time  of  conjunction, 
Geocentric  longitude  of  the  Sun  and  Mercury, 

Middle  apparent  time, 
Semi-duration  of  the  transit, 
Nearest  approach  of  centres, 

November  llth  1861. 
Mean  time  of  conjunction, 
Geocentric  longitude  of  the  Sun  and  Mercury- 


Middle  apparent  time, 
Semi-duration  of  the  transit, 
Nwirett  approach  of  cemtret, 


23 


H    M.    8. 

7  47  54 

8   D.    M.    8. 

7  14  43    8 

H    M.    8. 

8  12  22 
2  33  53 
-      0    5  37  South. 


H    M.    8. 

7  54  18 

8    D.    M.    8. 
1    18      1    49 

H    M.    8. 

7  32  58 
-     3  22  33 

-08  58  South. 


IX    M.    8. 

1  37  43 

8    D.    M.    8. 

7  17   19  19 


H    M.    8. 

1  49  43 

2  41  33 

2  36  North. 


H    M.    8. 

19  20  13 

8   D.    M.    8. 

7  19  54  44 

H    M.    8. 

19  20  14 
2  0  23 
0  10  02N0rtfc, 


24 


Of  the  Solar  System. 

November  4th  1868. 


Stc.  2 


Mean  time  of  conjunction, 
Middle  apparent  time, 
Semi-duration  of  the  transit, 

Geocentric  longitude  of  (he  Sun  and  Mercury, 
Nearest  approach  of  centres, 

May  6th  1878, 

Mean  time  of  conjunction, 
Middle  apparent  time,  - 

Semi-duration  of  the  transit, 

Geocentric  longitude  of  the  Sun  and  Mercury, 
Nearest  approach  of  centres, 

November  7ih  1881. 


II    M.    B. 

18  43  45 

19  18  24 
1  45  21 

*    D.    M.    8. 

7  13     9  42 

12  20  South. 


H  af.  ««L 

6  38  30' 
6  55  14 
3  53  31 

8    D.    M.    S. 

1  16     3  50 

4  39  North. 


Mean  time  of  conjunction, 
Middle  apparent  time, 
Semi-duration  of  the  transit, 


H    M.    8. 

12  39  38 

12  59  33 

2  39    6 


Geocentric  longitude  of  the  Sun  and  Mercury, 
Nearest  approach  of  centres, 

May  9th  1891. 


S    D.    M.    8. 

7  15  46  57 

3  57  South, 


Mean  time  of  conjunction, 
Middle  apparent  time, 
Semi-duration  of  the  transit, 


II    M.    8. 

14  44  57 

14  13  46 

2  34  20 


Geocentric  longitude  of  the  Sun  and  Mercury, 
Nearest  approach  of  centres, 

November  10th  1894. 


8   D.    M.    8. 
1    19      9      1 

0  12  21  South. 


Mean  time  of  conjunction, 
Middle  apparent  time, 
Semi-duration  of  the  transit, 


H    M.    8. 

6  17  5 
6  36  29 
2  37  36 


Geocentric  longitude  of  the  Sun  and  Mercury, 
Nearest  approach  of  centre*, 


6  D.    M.    S. 

7  18  22    9 

4  SONorth, 


Sec.  2 


Of  the  Solar  System. 


TRANSITS  OF  VENUS  OVER  THE  SUN'S  DISK,  FROM  THE 
YEAR  1769  TO  THE  YEAR  2004  INCLUSIVE. 

Jwi*  3d  1769. 


Mean  time  of  conjunction, 

Middle  apparent  time, 

Duration  of  the  transit,  -  * 

Geocentric  longitude  of  the  Sun  and  Venus, 
Nearest  approach  of  centres, 

December  8tk  1874, 


H  ac.  s. 

9  58  34 
10  27     3 

5  59  46     . 

*  D.  fir.  s, 
2  13  27     8 

0  10  lONorth. 


Mean  time  of  conjunction, 
Middle  apparent  time, 
Duration  of  the  transit, 


Geocentric  longitude  of  the  Sun  and  Venus, 
Nearest  approach  of  centres, 

December  6th  1882. 


II    M.    S. 

16     8  24 

15  43  28 

4    9  22 

B   D.    M.    B. 

8  16  67  49 

0  13  51  North. 


Mean  time  of  conjunction, 
Middle  apparent  time, 
Duration  of  the  transit, 

Geoncentric  longitude  of  the  Sun  and  Venus, 
Nearest  approach  of  centres, 

June  7th  2004. 

Mean  time  of  conjunction, 
Middle  apparent  time, 
Duration  of  the  transit, 

Geocentric  longitude  of  the  Sun  and  Venus, 
Nearest  approach  of  centres, 

The  earth  is  the  next  planet  above 
Solar  Sytem  ;  it  is  95,000,000  of  miles 
and  performs  a  revolution  around  him, 
tice,  or  Equinox,  to  the  same  again, 
c 


H  M.    B. 

4  16  24 

4  49  42 

6  8  26 

S    D.  M.    8. 

8  14  29  14 

10  29  South, 


H    M.    B. 

-      20  51  24 

20  26  59 

5  29  40 

8    D.    M.    S. 

2  17  54  23 

11  19  South. 

Venus,  in  the 

from  the  Sun, 

from  any  Sols- 

in  365  days,  5 


25  Of  the  Solar  System.  Sec.  2 

hours  and  49  minutes :  but  from  any  fixed  star  to  the 
same  again,  in  365  days,  6  hours  and  9  minutes.  The 
former  being  the  length  of  the  tropical  year,  and  the 
latter  the  sidereal.  It  travels  at  the  rate  of  58,000 
miles  every  hour,  in  performing  its  annual  revolution. 
It  revolves  on  its  own  axis  from  West  to  East,  once  in 
24  hours.  Its  mean  diameter  as  seen  from  the  Sun, 
is  17  seconds  and  two  tenths  of  a  degree.  Which,  by 
calculation,  will  give  about  7,970  miles  for  its  diame- 
ter. The  form  of  the  earth  is  an  oblate  spheroid, 
whose  equatorial  axis  exceeds  its  polar  by  36  miles, 
and  is  surrounded  by  an  atmosphere  extending  45  miles 
above  its  surface. 

The  Seas,  and  unknown  parts  of  the  earth,  (by  a 
measurement  of  the  best  Maps,)  contain  160  millions, 
522  thousand,  and  26  square  miles.  The  inhabited 
parts  38  millions,  990  thousand,  569.  Europe  four 
millions,  456  thousand,  and  65.  Asia  10  millions,  568 
thousand,  823.  Africa  9  millions,  654  thousand,  807. 
America  14  millions,  110  thousand,  874  :  the  whole 
amounting  to  199  millions,  512  thousand,  595  ;  which 
is  the  number  of  square  miles,  on  the  whole  surface  of 
the  Globe  we  inhabit. 

The  Moon  is  not  a  planet,  but  only  a  satellite,  or  an 
attendant  of  the  earth,  performing  a  revolution  round 
it  in  29  days,  12  hours  and  44  minutes  ;  and  with  the 
earth,  is  carried  round  the  Sun  once  in  every  year. 

The  diameter  of  the  Moon  is  2,180  miles,  and  her 
mean  distance  from  the  earth's  centre,  is  estimated  at 
240,000  miles.  She  goes  around  her  orbit  in  27  days, 


Sec.  2  Of  the  Solar  System.  27 

7  hours  and  43  minutes ;  moving  about  2,290  miles  ev- 
ery hour,  and  performs  a  revolution  on  her  own  axis 
exactly  in  the  time  that  she  goes  round  the  earth ;  con- 
sequently the  same  side  of  her  is  continually  presented 
towards  the  earth,  and  the  length  of  her  day  and  night 
taken  together,  is  equal  to  a  lunar  month.  Her  mean 
apparent  diameter,  as  seen  from  the  earth,  is  31  minutes 
and  8  seconds  of  a  degree.  The  orbit  of  the  Moon, 
crosses  the  ecliptic  in  two  opposite  points,  called  the 
Moon's  nodes,  consequently  one  half  of  her  orbit  is 
above  the  ecliptic,  and  the  other  below ;  the  angle  of  its 
obliquity  is  5  degrees,  and  20  minutes. 

The  Moon  has  scarcely  any  .difference  of  seasons,  be- 
cause her  axis  is  nearly  perpendicular  to  the  ecliptic, 
and  consequently  the  Sun  never  removes  sensibly  from 
her  equator. 

The  earth  which  we  inhabit,  serves  as  a  satellite  to 
the  Moon,  waxing  and  waning  regularly,  but  appearing 
thirteen  times  as  large,  and  affording  her,  thirteen 
times  as  much  light  as  the  Moon  does  to  us.  When 
she  is  new  to  us,  the  earth  appears  full  to  her ;  and, 
when  she  is  in  her  first  quarter  as  seen  from  the  earth, 
the  earth  is  in  its  third  quarter  as  seen  from  the  Moon. 

The  Moon  is  an  opaque  globe,  like  the  earth,  and 
shines  only  by  reflecting  the  light  of  the  Sun  ;  there- 
fore whilst  that  half  of  her  which  is  towards  the  Sun, 
is  enlightened,  the  other  half  must  be  dark  and  in- 
visible. Hence  she  disappears  when  she  comes  between 
us  and  the  Sun ;  because  her  dark  side  is  then  to- 
wards us. 


28  Of  the  Solar  System.  Sec.  2 

The  planet  Mars  is  next  in  order,  being  the  first  a- 
bove  the  earth's  orbit.  His  distance  from  the  Sun  is 
computed  at  144,000,000  of  miles,  and  by  travelling  at 
the  rate  of  54,000  miles  every  hour,  he  goes  round  the 
Sun  in  686  of  our  days,  23  hours  and  30  minutes,  which 
is  the  length  of  his  year,  equal  to  667  and  3-4th  of  his 
days ;  and  every  day  and  night  together,  being  nearly  40 
minutes  longer  than  with  us.  His  diameter  is  computed 
at  4,189  miles,  and  by  his  diurnal  rotation,  the  inhab- 
itants at  his  equator  are  carried  528  miles  every  hour. 
The  Sun  appears  to  the  inhabitants  of  Mars,  nearly 
two-thirds  the  size  that  it  does  to  us. 

His  mean  apparent  diameter,  as  seen  from  the  earth 
is  27  seconds,  and  as  seen  from  the  Sun,  ten  seconds 
of  a  degree.  His  axis  is  inclined  to  his  orbit  59  de- 
grees and  22  minutes. 

To  the  inhabitants  of  the  planet  Mars,  our  Earth  and 
Moon  appear  like  two  Moons  ;  the  one  being  13  times 
as  large  as  the  other  ;  changing  places  with  each 
other,  and  appearing  sometimes  horned,  sometimes 
half  or  three-quarters  illuminated  but  never  full,  nor 
at  most  above  one  quarter  of  a  degree  from  each  oth- 
er ;  although  they  are  in  fact  240,000  miles  asunder. 

This  Earth  appears  almost  as  large  from  Mars  as 
Venus  does  to  us.  It  is  never  seen  above  48  degrees 
from  the  Sun,  at  that  planet.  Sometimes  it  appears  to 
pass  the  disk  of  the  Sun,  and  likewise  Mercury  and 
Venus.  But  Mercury  can  never  be  seen  from  Mars 
by  such  eyes  as  our's  (unless  assisted  by  proper  in- 
strument*,) and  Venus  will  be  as  seldom  seen  as  we 


Sec.  2  Of  the  Solar  System.  29 

see  Mercury.  Jupiter  and  Saturn  are  as  visible  to 
Mars  as  to  us.  His  axis  is  perpendicular  to  the  eclip- 
tic, and  his  inclination  to  it  is  one  degree  and  51  min- 
utes. The  planet  Mars  is  remarkable  for  the  redness 
of  its  light,  the  brightness  of  its  polar  regions,  and  the 
variety. of  spots  which  appear  upon  its  surface.  The 
atmosphere  of  this  planet,  which  Astronomers  have 
long  considered  to  be  of  an  extraordinary  height  and 
density,  is  the  cause  of  the  remarkable  redness  of  its 
appearance.  When  a  beam  of  white  light  passes 
through  any  medium,  its  color  inclines  to  redness,  in 
proportion  to  the  density  of  the  medium;  and  the  space 
through  which  it  has  travelled.  The  momen- 
tum of  the  red,  or  least  refrangible  rays  being  greater 
than  that  of  the  violet,  or  most  refrangible,  the  former 
will  make  their  way  through  the  resisting  medium, 
while  the  latter  are  either  reflected  or  absorbed.  The 
color  of  the  beam  therefore  when  it  reaches  the  eye, 
must  partake  of  the  color  of  the  least  refrangible  rays, 
and  must  consequently  increase  with  the  number  of 

those  of  the  violet,  which  have  been  obstructed. 

Hence  we  discover,  that  the  morning  and  evening 
clouds  are  beautifully  tinged  with  red,  that  the  Sun, 
Moon  and  Stars  appear  of  the  same  color,  when  near 
the  horizon,  and  that  every  luminous  object  seen 
through  a  mist,  is  of  a  ruddy  hue.  There  is  a  great 
difference  of  color  among  the  planets,  we  are  there- 
fore, (if  the  preceding  observations  be  correct,)  under 
the  necessity  of  concluding,  that  those  in  which  the 
red  color  predominates,  are  surrounded  with  the  most 


30  Of  the  Solar  System.  Sec.  2 

extensive  and  dense  atmospheres.  According  to  this 
idea,  the  atmosphere  of  Saturn,  must  be  the  next  to 
that  of  Mars,  in  density  and  extent. 

The  planet  Mars  is  an  oblate  spheroid,  whose  equa- 
torial diameter  is  to  the  polar  as  1,355  is  to  1,272,  or 
nearly  as  16  to  15.  This  remarkable  flattening  at  the 
poles  of  Mars,  probably  arises  from  the  great  variation 
in  the  density  of  his  different  parts. 

VESTA. 

Some  Astronomers  supposed  that  a  planet  existed 
between  the  orbits  of  Jupiter  and  Mars ;  judging  from 
the  regularity  observed  in  the  distances  of  the  former 
discovered  planets  from  the  Sun.  The  discovery  of 
Ceres  confirmed  this  conjecture,  but  the  opinion  which 
it  seemed  to  establish  respecting  the  harmony  of  the 
Solar  System,  appeared  to  be  completely  overturned, 
by  the  discovery  of  Pallas  and  Juno.  Dr.  Olders, 
however  imagined  that  these  small  celestial  bodies 
were  merely  the  fragments  of  a  larger  planet  which 
had  burst  asunder  by  some  internal  convulsion,  and 
that  several  more  might  yet  be  discovered  between  the 
orbits  of  Mars  and  Jupiter.  He  therefore  concluded 
that  though  the  orbits  of  all  these  fragments  might  be 
inclined  to  the  ecliptic,  yet  as  they  must  have  all  diver- 
ged from  the  same  point,  they  ought  to  have  two  com- 
mon points  of  reunion,  or  two  nodes  in  opposite  regions 
of  the  Heavens,  through  which  all  the  planetary  frag- 
ments must  sooner  or  later  pass.  One  of  these  nodes 


Sec.  2  Of  the  Solar  System.  31 

he  found  to  be  in  Virgo,  and  the  other  in  the  Whale, 
and  it  was  actually  in  the  latter  of  these  regions,  that 
Mr.  Harding  discovered  the  planet  Juno.  With  the 
intention  therefore  of  detecting  other  fragments  of  the 
supposed  planet,  Dr.  Olders  examined  thrice  every  year 
all  the  little  stars  in  the  opposite  constellations  of  the 
Virgin,  and  the  Whale,  till  his  labors  were  crowned 
with  success  on  the  29th  of  March,  1807,  by  the  dis- 
covery of  a  new  planet  in  the  constellation  Virgo,  to 
which  he  gave  the  appropriate  name  of  Vesta.  The 
planet  Vesta  is  the  next  above  Mars,  and  is  in  appear- 
ance of  the  fifth  or  sixth  magnitude,  and  may  be  seen 
in  a  clear  morning  by  the  naked  eye.  Its  light  is  more 
intense,  pure  and  white,  than  either  of  the  three  follow- 
ing Ceres,  Juno,  or  Pallas.  Its  distance  from  the  Sun 
is  computed  at  225  millions  of  miles,  and  its  diameter 
at  238  :  its  revolutions  have  not  hitherto  been  sufficient- 
ly ascertained. 

ON  JUNO. 

The  planet  Juno,  the  next  above  Vesta,  and  between 
the  orbits  of  Mars,  &  Jupiter  was  discovered  by  Dr.  Har- 
ding, at  the  Observatory  near  Bremen,  on  the  evening  of 
the  5th  of  September,  1804.  This  planet  is  of  a  reddish 
color,  and  is  free  from  that  nebulosity  which  surrounds 
Pallas.  It  is  distinguished  from  all  the  other  planets 
by  the  great  excentricity  of  its  orbit,  and  the  effecc  of  this 
is  so  extremely  sensible,  that  it  passes  over  that  half  of 
its  orbit  which  is  bisected  by  its  perihelion  in  half  the 


32  Of  the  Solar  System.  Sec.  2 

time  that  it  employs  in  describing  the  other  half,  which 
is  further  from  the  Sun-,  from  the  same  cause  its  great- 
est distance  from  the  Sun  is  douhle  the  least.  Tne  dif- 
ference between  the  two  being  about  127  millions  of 
miles.  Its  mean  distance  from  the  Sun  is  computed  at 
252  millions  of  miles,  and  performs  its  tropical  revolu- 
tion in  4  years  and  128  days.  Its  diameter  is  estimated 
at  1,425  miles,  and  its  apparent  diameter  as  seen  from 
the  Earth,  three  seconds  of  a  degree,  and  its  inclination 
of  orbit  twenty-one  degrees. 

ON  CERES. 

The  planet  Ceres  was  discovered  at  Palermo,  in  Si- 
cily, on  the  first  of  January,  1801,  by  M.  Riazzi,  an 
ingenious  observer,  who  has  since  distinguished  him- 
self by  his  Astronomical  labors.  It  was  however  again 
discovered  by  Dr.  Ciders,  on  the  first  of  January,  1807, 
nearly  in  the  same  place  where  it  was  expected  from 
the  calculations  of  Baron  Zach.  The  planet  Ceres  is 
of  a  ruddy  color,  and  appears  about  the  size  of  a  star 
of  the  8th  magnitude.  It  seems  to  be  surrounded  with 
a  large  dense  atmosphere  of  675  miles  high,  according 
to  the  calculations  of  Schroeter,  and  plainly  exhibits  a 
disk,  when  examined,  with  a  magnifying  power  of  200. 

Ceres  is  situated  between  the  orbits  of  Mars  and  Ju- 
piter. She  performs  her  revolution  round  the  Sun  in 
four  years,  7  months  and  ten  days  ;  and  her  mean  dis- 
tance is  estimated  at  263  millions  of  miles  from  that  lu- 
minary. The  observations  which  have  been  hitherto 


Sec.  2  Of  the  Solar  System.  S3 

made  upon  this  celestial  body,  do  not  appear  sufficient- 
ly correct  to  determine  its  magnitude  with  any  degree 
of  accuracy. 

ON  PALLAS. 

The  planet  Pallas  was  discovered  at  Bremen,  in 
Lower  Saxony,  on  the  evening  of  the  28th  of  March, 
1802,  by  Doctor  Olders,  the  same  active  Astronomer, 
who  re-discovered  Ceres.  It  is  situated  between  the  or- 
bits of  Mars  and  Jupiter,  and  is  nearly  of  the  same 
magnitude  and  distance  with  Ceres,  but  of  a  less  ruddy 
color.  It  is  seen  surrounded  with  a  nebulosity  of  al- 
most the  same  extent,  and  performs  its  annual  revolution 
in  nearly  the  same  period.  The  planet  Pallas  however 
is  distinguished  in  a  very  remarkable  manner  from  Ce- 
res, and  all  the  primary  planets,  by  the  immense  inclin- 
ation of  its  orbit.  While  these  bodies  are  revolving  round 
the  Sun  in  almost  circular  paths,  rising  only  a  few  de- 
grees above  the  plane  of  the  ecliptic  ;  Pallas  ascends  a- 
bove  this  plane,  at  an  angle  of  about  35  degrees,  which 
is  nearly  five  times  greater  than  the  inclination  of  Mer- 
cury. From  the  eccentricity  of  Pallas  being  greater 
than  that  of  Ceres,  or  from  a  difference  of  position  in  the 
line  of  their  apsides,  where  their  mean  distances  are 
nearly  equal,  the  orbits  of  these  two  planets  mutually 
intersect  each  other  ;  a  phenomenon  which  is  altogether 
anomalous  in  the  Solar  System. 

Pallas  performs  its  tropical  revolution  in  feur  years  7 
months  and  11  days.  The  distance  of  this  planet,  from 


34  Of  the  Solar  System.  Sec.  2 

the  Sun,  is  estimated  at  265  millions  of  miles.      It  is 
surrounded  with  an  atmosphere  468  miles  high. 

OF  JUPITER, 

Jupiter,  the  largest  of  all  the  planets,  is  still  higher  in 
the  Solar  System,  being  four  hundred  and  ninety  mil- 
lions of  miles  from  the  Sun,  and  by  performing  his  an- 
nual revolution  round  the  Sun  in  eleven  years,  314 
days,  20  hours  and  27  minutes,  he  moves  in  his  orbit 
at  the  rate  of  29,000  miles  in  an  hour.  The  diameter 
of  this  planet  is  estimated  at  89,170  miles,  and  performs 
a  revolution  on  its  own  axis  in  nine  hours,  55  minutes 
and  37  seconds  ;  which  is  more  than  28,000  miles  every 
hour,  at  his  equator,  the  velocity  of  motion  on  his  axis 
being  nearly  equal  to  the  velocity  with  which  he  moves 
in  his  annual  orbit. 

This  planet  is  surrounded  by  faint  substances  called 
belts,  in  which  so  many  changes  appear,  that  they  have 
been  regarded  by  some,  as  clouds  or  openings  in  the  at- 
mosphere of  the  planet ;  while  others  imagine  that  they 
are  of  a  more  permanent  nature,  and  are  the  marks  of 
great  physical  revolutions  which  are  perpetually  chang- 
ing the  surface  of  the  planet. 

The  axis  of  Jupiter  is  so  nearly  perpendicular  to  his 
orbit,  that  he  has  no  sensible  change  of  seasons,  which 
is  a  great  advantage,  and  wisely  ordered  by  the  Author 
of  nature;  for  if  the  axis  of  this  planet  were  inclined  any 
considerable  number  of  degrees,  just  so  many  degrees 
round  each  pole  would  in  their  turn,  be  almost  six  of 


Sec.  2  Of  the  Solar  System.  35 

our  years  together  in  darkness,  and,  as  each  degree  of  a 
great  circle  on  Jupiter  contains  778  of  our  miles  at  a 
mean  rate  ;  judge  ye  what  vast  tracts  of  lands  would  be 
rendered  uninhabitable  by  any  considerable  inclination 
of  his  axis. 

The  difference  between  the  equatorial  and  polar  di- 
ameters of  this  oblate  spheroid  is  .computed  at  6,230 
miles  ;  for  his  equatorial  diameter  is  to  his  polar,  as!3 
is  to  12;  consequently  his  poles  are  3,115  miles  nearer 
his  centre  than  his  equator.  This  results  from  his  rapid 
motion  round  his  axis,  for  the  fluids  together  with  the 
light  particles  which  they  can  carry,  or  wash  away  with 
them,  recede  from  the  poles,  which  are  at  rest  towards 
the  equator,  where  the  motion  is  more  rapid,  until  there 
be  a  sufficient  number  of  such  particles  accumulated  to 
make  up  the  deficiency  of  gravity  occasioned  by  the  cen- 
trifugal force,  which  arises  from  a  quick  motion  round 
an  axis  j  and  when  the  deficiency  of  weight  or  gravity 
of  the  particles  is  made  up  by  a  sufficient  accumulation, 
the  equilibrium  is  restored,  and  the  equatorial  parts  rise 
no  higher.  The  orbit  of  Jupiter  is  inclined  to  the  eclip- 
tic one  degree  and  20  minutes.  His  north  node  is  in 
the  7th  degree  of  Cancer,  and  his  south  node  in  the  7th 
degree  of  Capricorn.  His  mean  apparent  diameter  as 
seen  from  the  earth  is  39  seconds,  and  as  seen  from  the 
Sun,  37  seconds  of  a  degree. 

This  planet  being  situated  at  so  great  a  distance  from 
the  Sun,  does  not  enjoy  that  degree  of  light  emanating 
from  his  rays,  which  is  enjoyed  by  the  earth.  To  sup- 
ply this  deficiency,  the  great  Author  of  our  existence  has 


36  Of  the  Solar  System.  Sec.  8 

provided  4  satellites,  or  Moons  to  be  his  constant  atten- 
dants, which  revolve  around  him,  in  such  manner,  that 
scarcely  any  part  of  this  large  planet  but  is  enlightened 
during  the  whole  night,  by  one  or  more  of  these  Moons* 
except  at  his  poles,  where  only  the  farthest  Moons  can 
be  seen  ;  there,  however  this  light  is  not  wanted  ;  be- 
cause the  Sun  constantly  circulates  in  or  near  the  hori- 
zon, and  is  very  probably  kept  in  view  of  both  poles  by 
the  refraction  of  his  atmosphere.  The  first  Moon,  or 
that  nearest  to  Jupiter  performs  a  revolution  around  him 
in  one  day,  18  hours  and  36  minutes  of  our  time,  and  is 
229  thousand  miles  distant  from  his  centre  :  the  second 
performs  his  revolution  in  3  days  13  hours  and  15 
minutes  at  a  distance  of  364  thousand  miles  :  the  third 
in  seven  days,  three  hours  and  69  minutes,  at  the  dis- 
tance of  580  thousand  miles,  and  the  fourth,  or  farthest 
from  his  centre  in  16  days,  18  hours,  and  30  minutes, 
at  the  distance  of  one  million  of  miles  from  his  centre. 
The  angles  under  which  these  satellites  are  seen  from 
the  earth,  at  its  mean  distance  from  Jupiter,  are  as  fol- 
lows : —  The  first  three  minutes  and  65  seconds :  the 
second  six  minutes  and  15  seconds :  the  third  9  minutes 
and  58  seconds,  and  the  fourth  17  minutes  and  30  sec- 
onds. This  planet  when  seen  from  its  nearest  Moon, 
must  appear  more  than  one  thousand  times  as  large  as 
our  Moon  does  to  us. 

The  threl  nearest  Moons  to  Jupiter,  pass  through  his 
shadow  and  are  eclipsed  by  him,  in  every  revolution,  but 
the  omt  oi  the  fourth  is  so  much  inclined,  that  it  passes 
by  itsopposition  to  Jupiter  without  entering  his  shadow, 


2  Of  th*  Solar  System.  37 

two  years  in  every  six.  By  these  eclipses,  Astronomers 
have  not  only  discovered  that  the  Sun's  light  is  about  8 
minutes  in  coming  to  us ;  but  they  have  also  determ:n:d 
the  longitude  of  places  on  this  earth  with  greater  certain- 
ty, and  facility  than  by  any  other  method  yet  known. 

OF  SATURN. 

Saturn  is  the  most  remarkable  of  all  the  planets,  it  is 
calculated  at  9  hundred  millions  of  miles  from  the  Sun, 
and  travelling  at  the  rate  of  21,900  miles  every  hour, 
and  performs  its  annual  circuit  in  29  years,  167  days 
and  2  hours  of  our  time  ;  which  makes  only  one  year 
to  that  planet.  Its  diameter  is  computed  at  79,042 
miles,  and  performs  a  revolution  on  its  own  axis  once 
in  ten  hours,  16  minutes  and  two  seconds.  Its  mean 
apparent  diameter  as  seen  from  the  earth,  is  18  seconds, 
and  as  seen  from  the  Sun,  16  seconds  of  a  degree;  its 
axis  is  supposed  to  be  60  degrees  inclined  to  its  orbit. 

This  planet  is  surrounded  by  a  thin  broad  ring,  which 
no  where  touches  its  body,  and  when  viewed  by  the 
aid  of  a  good  telescope  appears  double.  It  is  inclined 
30  degrees  to  the  ecliptic,  and  is  about  21  thousand 
miles  in  breadth ;  which  is  equal  to  its  distance  from 
Saturn  on  all  sides.  This  ring  performs  a  revolution 
on  its  axis  in  the  same  space  of  time  with  the  planet, 

namely,  ten  hours  16  miutes  and  two  seconds. 

This  ring  seen  from  the  planet  Saturn,  appears  like  a 
vast  luminous  circle  in  the  Heavens,  and,  as  if  ft  does 
not  belong  to  the  planet  When  we  see  the  ring  most 


38  Of  the  Solar  System.  Sec  2. 

open,  its  shadow  upon  the  planet  is  broadest,  and  from 
that  time  the  shadow  grows  narrower,  as  the  ring  ap- 
pears to  do  to  us,  until  by  Saturn's  annual  motion,  the 
Sun  comes  to  the  plane  ojf  the  ring,  or  even  with  its 
edge ;  which  being  then  directed  towards  the  earth,  it 

becomes  to  us  invisible  on  account  of  its  thinness. 

The  ring  nearly  disappears  twice  in  every  annual  revo- 
lution of  Saturn,  when  he  is  in  the  19th  degree,  both 
of  Pisces  and  Virgo.  But,  when  Saturn  is  in  the  19th 
degree  either  of  Gemini  or  Sagitarius,  his  ring  appears 
most  open  to  us,  and  then  its  longest  diameter  is  to  its 
shortest  as  9  to  4. 

This  planet  is  surrounded  with  no  less  than  seven 
satellites,  which  supply  him  with  light  during  the  ab- 
sence of  the  Sun.  The  fourth  of  these  was  first  dis- 
covered by  Huygens,  on  the  25th  of  March,  1655. 

Cassini  discovered  the  fifth  in  October,  1671.  The 
third  on  the  23d  of  december,  1672 :  And  the  first  and 
second  in  the  month  of  March,  1684.  The  sixth  and 
seventh  were  discovered  by  Dr.  Herschel  in  the 
year  1789.  These  are  nearer  to  Saturn  than  any 
of  the  others. 

These  Moons  perform  their  revolutions  round  this 
planet  on  the  outside  of  his  ring,  and  nearly  in  the 
same  plane  with  it.  The  first,  or  nearest  Moon  to 
Saturn,  performs  its  periodical  revolution  around  him 
in  22  hours  and  37  minutes,  at  the  distance  of  121,000 
miles  from  his  centre  ;  the  second  performs  its  period- 
ical revolution  in  one  day,  8  hours  and  53  minutes,  at 
the  distance  of  156  thousand  miles  ;  the  third  in  one 


Sec.  2  Of  the  Solar  System  39 

day,  21  hours,  18  minutes  and  26  seconds,  at  the  dis- 
tance of  193  thousand  ;  the  fourth  in  two  days,  17  hours 
44  minutes  and  51  seconds,  at  the  distance  of  247  thou- 
sand ;  the  fifth  in  4  days,  12  hours,  25  minutes  and  1 1 
seconds,  at  346  thousand ;  the  6th  in  15  days,  22  hours, 
41  minutes  and  13  seconds,  at  the  distance  of  802  thou- 
sand ;  and  the  7th  or  outermost  in  49  days,  7  hours,  53 
minutes  and  43  seconds,  at  the  distance  of  two  millions, 
337  thousand  miles  from  the  centre  of  Saturn — their 
primary. 

When  we  look  with  a  good  telescope,  at  the  body  of 
Saturn,  he  appears  like  most  of  the  other  planets,  in  the 
form  of  an  oblate  spheroid,  arising  from  the  rapid  rota- 
tion about  his  axis.  He  however  appears  more  flat- 
tened at  the  poles,  than  any  of  the  others,  and  although 
his  motion  on  his  axis  is  not  equal  to  that  of  Jupiter, 
yet  he  does  not  appear  to  be  in  form,  so  near  that  of  a 
globe  as  that  planet.  When  we  consider  that  the  ring 
by  which  Saturn  is  encompassed,  lies  in  the  same  plane 
of  his  equator,  and,  that  it  is  at  least  equal  if  not  more 
dense  than  the  planet,  we  shall  find  no  difficulty  in  ac- 
counting for  the  great  accumulation  of  matter,  at  the 
the  equator  of  Saturn.  The  ring  acts  more  powerful- 
ly upon  the  equatorial  regions  of  Saturn;  than  upon  any 
part  of  his  disk  ;  and  by  diminishing  the  gravity  of 
these  parts,  it  aids  the  centrifugal  force  in  flattening 
the  poles  of  the  planet.  Had  Saturn  indeed  never  re- 
volved upon  his  axis,  the  action  of  the  ring  would  of 
itself  have  been  sufficient,  to  have  given  it  the  form  of  a 
spheroid. 


40  Of  the  Solar  System.  Sec.  S 

The  following,  are  the  dimensions  of  this  luminous 
zone,  as  determined  by  Dr.  Herschel. 
Inside  diameter  of  the  interior  ring,       -  146,345 

Outside  diameter  of  the  interior  ring,  -;>  184,383 
Inside  diameter  of  the  exterior  ring,  *  190,240 
Outside  diameter  of  the  exterior  ring,  .*  204,833 
Breadth  of  the  interior  ring,  ...  20,000 
Breadth  of  the  exterior  ring,  -  -  '-  7,200 
Breadth  of  the  dark  space  between  the  two  rings,  2,839 
Angle  which  it  subtends  when  seen  at  the  mean  M.  s. 

distance  of  the  planet.         -         -         -         -      7  25 

ON  HERSCHEL,  OR  URANUS. 

From  inequalities  in  the  motion  of  Jupiter  and  Saturn, 
for  which  no  rational  account  could  be  given,  and  from 
the  mutual  action  of  these  planets,  it  was  inferred  by 
some  Astronomers,  that  another  planet  existed  beyond 
the  orbits  of  Jupiter  and  Saturn ;  by  whose  action  these 
irregularities  were  produced.  This  conjecture  was 
confirmed  on  the  13th  of  March,  1781 ;  when  Dr.  Her- 
schel discovered  a  new  planet,  which  in  compliment  to 
his  Royal  Patron,  he  called  Georgium  Sidus,  although 
it  is  more  generally  known  by  the  name  of  Herschel, 
or  Uranus.  This  new  planet,  (which  had  been  former- 
ly observed  as  a  small  star  by  Flamstead,  and  likewise 
by  Tobias  Mayer,  and  introduced  into  their  catalogue 
of  fixed  stars,)  is  situated,  one  thousand,  eight  hundred 
millions  of  miles  from  the  centre  of  the  System,  and 
performs  its  revolution  round  the  Sun  in  83  years,  150 


Sec.  2  Of  the  Solar  System.  41 

days,  and  18  hours.  Its  diameter  is  computed  at 
35,112  miles.  When  seen  from  the  earth,  its  mean 
apparent  diameter  is  three  and  J  seconds,  and  as  seen 
from  the  Sun,  4  seconds  of  a  degree.  As  the  distance 
of  this  planet  from  the  Sun  is  twice  as  great  as  that  of 
Saturn,  it  can  scarcely  be  distinguished  without  the 
aid  of  instruments.  When  the  sky  however  is  serene 
it  appears  like  a  fixed  star  of  the  sixth  magnitude,  with 
a  bluish  white  light,  and  a  brilliancy  between  that  of 
Venus  and  the  Moon  ;  but  seen  with  a  power  of  two 
or  three  hundred,  its  disk  is  visible  and  well  defined.— 
The  want  of  light  arising  from  the  distance  of  this 
planet  from  the  Sun,  is  supplied  by  six  satellites,  all  of 
which  were  discovered  by  Dr.  Herschel. 

The  first  of  those  satellites  is  twenty  five  and  half 
seconds  from  its  primary,  and  revolves  round  it  in  5 
days,  21  hours  and  25  minutes ;  the  second  is  nearly  34 
seconds  distant  from  the  planet,  and  performs  its  rev- 
olution in  8  days,  17  hours,  1  minute  and  19  seconds. 
The  distance  of  the  third  satellite  is  38,57  seconds, 
and  the  time  of  its  periodical  revolution  is  ten  days,  23 
hours,  and  four  minutes.  The  distance  of-the  fourth 
satellite  is  44,22  seconds,  and  the  time  of  its  periodical 
revolution  is  13  days,  11  hours,  5  minutes  and  30  sec- 
onds. The  distance  of  the  fifth  is  one  minute  and  28 
seconds,  and  its  revolution  is  completed  in  38  days,  1 
hour,  and  49  minutes.  The  sixth  satellite,  or  the  fur- 
thest from  the  centre  of  its  primary,  at  the  distance  of 
two  minutes,  and  nearly  57  seconds,  and  therefore  re- 
quires 107  days,  16  hours,  and  40  minutes  to  complete 


42  Of  the  Solar  System.  '  Sec.  2 

one  revolution.  The  second  and  fourth  of  these  were 
discovered  on  the  llth  of  January,  1787,  the  other  four 
were  discovered  in  1 790,  and  1 794 ;  but  their  distan- 
tances,  and  times  of  periodical  revolution,  have  not 
been  so  accurately  ascertained  as  the^  other  two.  It  is 
however,  a  remarkable  circumstance,  that  the  whole 
of  these  satellites  move  in  a  retrogade  direction,  and  in 
orbits  lying  in  the  same  plane,  and  almost  perpendicu- 
lar to  the  ecliptic. 

When  the  Earth  is  in  its  perihelion,  and  Herschel  in 
its  aphelion,  the  latter  becomes  stationary,  as  seen  from 
the  Earth,  when  his  elongation,  or  distance  from  the 
Sun  is  8  signs,  17  degrees,  and  37  minutes,  his  retro- 
gradations  continue  151  days,  and  12  hours.  When  the 
Earth  is  in  its  aphelion,  and  Herschel  in  its  perihelion, 
it  becomes  stationary,  at  an  elongation  of  8  signs,  16 
degrees,  and  27  minutes,  and  its  retrogadations  con- 
tinue 149  days  and  18  hours. 

ON  COMETS. 

Comets  are  a  class  of  celestial  bodies,  which  occa- 
sionly  appear  in  the  Heavens.  They  exhibit  no  visible 
or  defined  disk,  but  shine  with  a  pale  and  cloudy  light, 
accompanied  with  a  tail  or  train,  turned  from  the  Sun. 
They  traverse  every  part  of  the  Heavens,  and  move 
in  every  possible  direction. 

When  examined  through  a  good  telescope,  a  Comet 
resembles  a  mass  of  aquious  vapors,  encircling  an 
opaque  nucleus,  of  different  degrees  of  darknes  in  dif- 


Sec.  2  Of  the  Solar  System  43 

ferent  Comets  ;  though  sometimes,  as  in  the  case  of  sev- 
eral discovered  by  Dr.  Herschel,  no  nucleus  can 
be  seen. 

As  the  Comet  advances  towards  the  Sun,  its  faint 
and  nebulous  light  becomes  more  brilliant,  and  its  lu- 
minous train  gradually  increases  in  length. 

When  it  reaches  its  perihelion,  the  intensity  of  its 
light,  and  the  length  of  its  tail  reaches  their  maximum, 
and  then  it  sometimes  shines  with  all  the  splendor  of 
Venus.  During  its  retreat  from  the  perihelion,it  is  shorn 
of  its  splendor,  and  it  gradually  resumes  its  nebulous 
appearance  ;  and  its  train  decreases  in  magnitude,  until 
it  reaches  such  a  distance  from  the  Earth,  that  the 
attenuated  light  of  the  Sun,  which  it  reflects,  ceases  to 
make  an  impression  on  the  organ  of  sight.  Traversing 
unseen  the  remote  portion  of  its  orbit,  the  Comet  wheels 
its  etherial  course  far  beyond  the  limits  of  the  Solar 
System.  What  region  it  there  visits,  or  Upon  what 
destination  it  is  sent,  the  limited  powers  of  man  are  un- 
able to  discover.  After  the  lapse  of  years,  wre  per- 
ceive it  again  returning  to  our  System,  and  tracing  a 
portion  of  the  same  orbit  round  the  Sun,  which  it  had 
formerly  described. 

Various  opinions  have  been  entertained  by  Astrono- 
mers respecting  the  tails  of  Comets.  These  tails  or 
trains,  sometimes  occupy  an  immense  space  in  the 
Heavens.  The  Comet  of  1681,  stretched  its  tail  across 
an  arch  of  104  degrees ;  and  the  tail  of  the  Comet  of 
1769  subtended  an  angle  of  60  degrees  at  Paris,  70  at 
Bologna,  97  at  the  Isle  of  Bourbon,  and  90  degrees  at 


44  Of  the  Solar  System.  Sec  2. 

Sea,  between  Teneriffe  and  Cadiz.  These  long  trains 
of  light  are  maintained  by  Newton,  to  be  a  thin  vapor, 
raised  by  the  heat  of  the  Sun  from  the  Comet. 

If  we  knew  their  uses  in  our  System,  we  could  form 
more  probable  conjectures  as  to  the  chronology  of  their 
creation.     They  have  been  noticed  from  the  earliest 
era  of  our   Astronomical  History,  and  if  our  modern 
Philosophers  had  not  discovered,  that  some  (at  least,) 
leave  us  to  return  again  into  our  System,  and  there- 
fore describe  a  vast  elliptical  orbit  round  our  Sun,  we 
might  have  fancied  that  the  periods  of  their  first  recor- 
ded appearances  in  our  field  of  science,  were  the  eras 
of  their   individual  formation.      But   their   recurring 
presence  proves,  that  their  first  existence  ascends  in- 
to unexplored  and   unrecorded  antiquity.     Yet,  from 
whence  they  came  to  us,  we  as  little  know  as  for  for 
what  purpose.     Tycha  Brache  proved  that  they  were 
further  from  the   Earth  than  the   Moon,  and  were 
nearly  as  distant  as  the  planets.      The  Comet  of  1682, 
re-appeared  in  1 759,  in  the  interval  described,  an  orbit 
in  the  form  of  an  ellipsis,  answering  to  a  revolution  of 
27,937  days.      It  will  therefore  re-appear  in  Novem- 
ber, 1835.     In  its  greatest  distance,  it  is  supposed  not 
to  go  above  twice  as  far  as  Uranus.     This  is  indeed  a 
prodigious  sweep  of  space,  and  it  has  been  justly  obser- 
ved, that  the  vast  distance  to  which  some   Comets 
roam,  proves  how*  very  far  the  attraction  of  the  Sun 
extends ;  for  though  they  stretch  themselves  to  such 
depths  in  the  abyss  of  space,  yet  by  virtue  of  the  Solar 
power,  they  return  into  its  effulgence.     But  it  ha«  been 


Sec.  2  Of  the  Solar  System.  45 

recently  discovered,  that  three  Comets  (at  least,)  never 
leave  the  planetary  system.  One  whose  period  is  three 
years  and  a  quarter,  is  included  within  the  orbit  of  Ju- 
piter ;  another  of  six  years  and  three  quarters,  ex- 
tends not  so  far  as  Saturn  ;  and  a  third  of  twenty  years* 
is  found  not  to  pass  beyond  the  circuit  of  Uranus. 

The  transient  effect  of  a  Comet  passing  near  the 
Earth,  could  scarcely  amount  to  any  great  convulsion, 
but  if  the  Earth  were  actually  to  receive  a  shock  by 
collision,  from  one  of  those  bodies,  the  consequences 
would  be  awful.  A  new  direction  would  be  given  to 
its  rotatory  motion,  and  the  Globe  would  revolve  around 
a  new  axis.  The  Seas,  forsaking  their  ancient  beds, 
would  be  hurried  by  their  centrifugal  force  to  the  new 
equatorial  regions ;  islands  and  continents,  the  abodes 
of  men  and  animals,  would  be  covered  by  the  univer- 
sal rush  of  the  waters  to  the  new  Equator,  and  every 
vestige  of  human  industry  and  genius  at  once  de- 
stroyed. 

Although  the  orbits  of  all  the  planets  in  the  Solar 
System  be  crossed  by  five  hundred  different  Comets, 
the  chances  against  such  an  event  however,  are  so  ve- 
ry numerous,  that  there  need  be  no  dread  of  its  occur- 
rence ;  besides,  that  Almighty  arm  which  first  created 
them,  and  described  for  them  their  various  orbits — 
that  Omnipotent  Wisdom  which  directed  the  times  of 
their  periodical  revolutions,  still  continues  to  guide  and 
protect  all  the  workmanship  of  his  hands. 


46  Interrogations  for  Section  Second.  Sec.  2 

Interrogations  for  Section  Second. 

Of  what  does  the  SOLAR  SYSTEM  consist  1 

What  planets  finish  their  circuits  soonest  1 

Which  moves  with  the  greatest  rapidity  ? 

In  what  direction  do  they  move  in  their  orbits  ? 

What  is  jthe  form  of  the  orbits  described  by  the 
planets  ? 

Where  is  the  Sun  placed  7 

In  what  time  does  he  turn  round  on  his  own  axis  ? 

How  is  that  proved  1- 

What  is  his  diameter  1 

What  is  his  mean  apparent  diameter  as  seen  from 
the  Earth  1 

How  is  his  solidity  calculated  ? 

What  is  the  Ecliptic  ? 

What  is  meant  by  the  Nodes  ? 

Which  is  the  Ascending  Node  ? 

Which  is  the  Descending  Node  1 

How  many  signs  are  they  asunder  ? 

What  additional  Astronomical  discoveries  were 
made  in  the  year  1805  ? 

What  planet  is  nearest  the  Sun  1 

What  reasons  are  given  to  suppose  that^this  planet 
receives  its  light  from  the  Sun  ? 

What  is  the  computed  distance  of  Mercury  from  the 
Sun? 

What  is  its  diameter  ? 


Sec.  2.  Interrogations  for  Section  Second.  47 

In  what  time  does  it  perform  a  revolution  around  the 
Sun  ? 

What  time  on  its  own  its  axis  1 

How  many  miles  does  it  move  in  an  hour  in  its  mo- 
tion round  the  Sun  ? 

How  many  degrees  is  his  orbit  inclined  towards  the 
Ecliptic  ? 

What  planet  is  next  to  Mercury  ? 

What  is  the  distance  of  this  planet  from  the  Sun  ?  , 

How  many  miles  in  an  hour,  does  Venus  move  in 
performing  her  revolution  round  the  Sun  ? 

In  what  time  does  she  perform  her  annual  revolution  ? 

In  what  time  does  she  perform  a  revolution  on  her 
axis  ? 

What  is  her  diameter  ? 

How  is  it  known  that  the  orbits  of  Mercury  and  Ve- 
nus are  included  within  that  of  the  Earth  ? 

How  many  degrees  at  most  does  Mercury  depart 
from  the  Sun  ? 

How  many  Venus  ? 

What  is  meant  by  inferior  conjunction  ? 

What  is  a  transit  ?  % 

What  is  the  name  of  the  third  planet  from  the  Sun  ? 

What  is  its  distance  from  the  Sun  ? 

What  is  its  diameter  ? 

In  what  time  does  it  perform  a  revolution  around  the 
Sun? 

What  is  its  hourly  progress  1 

In  what  time  does  it  perform  a  revolution  on  its  axis  ? 

What  is  its  form  ? 


48  Interrogations  for  Section  Second.  Sec.  2 

How  many  miles  difference  in  the  two  diameters  ? 

What  is  the  Moon  1 

In  what  time  does  she  perform  a  revolution  round  the 
Sun  ? 

Around  the  Earth  ? 

Around  her  own  axis  ? 

What  is  her  distance  from  the*  Earth  ? 

How  many  miles  in  diameter  ? 

What  is  her  mean   apparent  diameter  as  seen  from 
the  Earth  ? 

Do  the  orbits  of  the  Earth  and  Moon  coincide  ? 

How  much   more  light  does  the  Earth   give  to  the 
Moon,  than  the  Moon  gives  to  us  ? 

What  is  the  name  of  the  fourth  planet  from  the  Sun  ? 

What  is  his  distance  from  that  luminary  ? 

In  what  time  does  he  perform  his  annual  revolution  ? 

What  is  his  hourly  progress  ? 
•  What  time  his  revolution  on  his  axis  ? 

What  is  his  mean  apparent  diameter  as  seen  from  the 
Earth  ? 

What  as  seen  from  the  Sun  ? 

For  what  is  the  planet  Mars  remarkable  ? 

What  have  Astronomers  concluded  to  be  the  cause  of 
this  remarkable  appearance  ? 

What  is  its  form  ? 

What  is  the  name  of  the  fifth  planet  from  the  Sun  ? 

By  whom  was  it  discovered  ?  And  when  ? 

In  what  sigh  of  the  Ecliptic  can  this  planet  be  seen 
without  the  aid  of  a  telescope  ? 

What  is  its  distance  from  the  Sun  ? 


Sec.  2          Interrogation*  for  Section  Second  49 

What  is  its  diameter  ? 

What  is  the  name  of  the  sixth  ? 

By  whom  was  it  discovered  1   And  when  ? 
What  is  its  color  ? 

For  what  is  it  distinguished  ? 

What  is  its  distance  from  the  Sun  ? 

What  is  the  time  of  its  tropical  revolution  7 

What  is  its  diameter  ? 

What  is  its  apparent  diameter  as  seen  from  the  Sun? 

What  is  the  name  of  the  seventh  ? 

By  whom  was  it  first  discovered  ? 

In  what  year  ? 

What  is  the  height  of  its  atmosphere  1 

In  what  time  does  this  planet  perform  its  revolution 
round  the  Sun  ? 

What  is  her  distance  from  that  luminary  ? 

What  is  the  name  of  the  eighth  planet  from  the  Sun  ? 

When  was  it  discovered  ?  And  by  whom  1 

What  is  its  distance  from  the  Sun  1 

In  what  time  does  it  perform  its  annual    revolution 
around  him  ? 

What  is  the  height  of  its  atmosphere  ? 

By  what  name  are  the  last  four  collectively  called  ? 

They  are  called  Asteroids. 

What  is  the  name  of  the  ninth  planet  from  the  Sun  ? 

How  many  miles  distant  from  the  Sun  ? 

What  is  his  diameter  1 

In  what  time  does  he  perform  his  annual  revolution  ? 

In  what  time  on  his  own  axis  ? 

What  is  his  mean  motion  in  his  orbit  1 


50  Interrogations  for  Section  Second.  Sec.  2 

What  is  his  mean  motion  on  his  axis  1 

Have  the  inhabitants  of  Jupiter  any  sensible  change 
of  seasons  ? 

How  many  miles  constitute  a  degree  on  this  planet  1 

How  many  miles  difference  between  his  equatorial 
and  polar  diameters  1 

Why  is  this  great  difference  1 

How  many  degrees  is  the  orbit  of  Jupiter  inclined  to 
the  Ecliptic  1  In  what  sign  of  the  Zodiac  is  his  north 
node  ?  In  what  sign  his  south  node  1 

How  many  satellites  attend  this  planet  7  What  is 
his  apparent  diameter  as  seen  from  the  Sun  ?  What 
as  seen  from  the  Earth  1 

Of  what  benefit  have  those  Moons  been  to  the  inhab- 
itants of  this  Earth  ?  Can  either  of  them  be  seen  by  us 
without  the  aid  of  telescopes  ? 

What  name  is  given  to  the  next,  or  tenth  planet 
from  the  Sun  ? 

What  is  its  distance  from  the  Sun  ? 

How  many  miles  does  this  planet  move  in  an  hour  ? 

In  what  time  does  it  perform  its  revolution  around 
the  Sun  ?  In  what  time  on  its  own  axis  1 

What  is  its  diameter  ? 

What  is  his  mean  apparent  diameter  as  seen  from  the 
Sun  1  What  as  seen  from  the  Earth  ? 

How  many  degrees  is  its  axis  inclined  to  its  orbit  ? 

What  encircles  his  body  1 

How  does  it  appear  when  viewed  with  a  telescope  1 

What  is  its  breadth  1 

How  does  it  appear  to  the  inhabitants  of  Saturn  1 


Sec.  2.  Interrogations  for  Section  Second.  5 1 

Why  is  it  sometimes  invisible  to  us  1 

How  many  times  does  it  appear  in  one  revolution  of 
the  planet  ? 

In  what  signs  and  degrees  of  the  Ecliptic  does  it 
disappear  1 

In  what  signs  and  degrees  does  this  ring  appear  most 
open  ? 

How  many  Satellites  has  this  planet  1 

Where   are   they  situated,  inside  or  outside  of  the 


ring? 


What  is  the  form  of  this  planet  1 

What  is  the  name  of  the  next,  or  outermost  planet  ? 

When  was  it  discovered  1 

How  far  is  it  situated  from  the  Sun  1 

In  what  time  does  it  perform  its  annual  revolution  1 
What  is  its  diameter  ? 

What  is  its  apparent  diameter  as  seen  from  the  Sun  ? 
What  as  seen  from  the  Earth  ? 

Can  it  be  seen  without  the  aid  of  a  telescope  ? 

How  many  Satellites  attend  it  ? 

In  what  direction  do  those  Satellites  move  ? 

What  are  Cornets  ? 

In  what  direction  do  they  move  1 

In  what  part  of  its  orbit  is  its  train  most  brilliant  ? 

What  was  Newton's  opinion  concerning  the  Comet's 
tail,  or  train  ? 

How   many  Comets  are  supposed  to  belong  to  the 
Solar  System  ? 


52  Interrogations  for  Section  Second.  Ssc.  2 

How  many  are  known  not  to  exceed  the  circuit  o  f 
Uranus  ? 

What  would  be  the  result  if  a  Comet  should  come  in 
actual  collision  with  this  Earth  ? 

Is  it  probable  that  there  will  ever  be  such  an  oc- 
currence ? 

Why  is  it  not  probable  ? 


SECTION  THIRD. 


THE  power  by  which  bodies  fall  towards  the  Earth, 
is  Called  GRAVITY,  or  Attraction.  By  this  power  in 
the  Earth,  it  is  that  all  the  bodies  on  whatever  side^ 
fall  in  lines  perpendicular  to  its  surface.  On  opposite 
parts  of  the  Earth,  bodies  fall  in  opposite  directions,  all 
towards  the  centre,  where  the  whole  force  of  gravity 
appears  to  be  accumulated.  By  this  power  constantly 
acting  on  bodies  near  the  Earth,  they  are  kept  from 
leaving  it,  and  those  on  its  surface  are  kept  by  it,  that 
they  cannot  fall  from  it.  Bodies  thrown  with  any  ob- 
liquity, are  drawn  by  this  power  from  a  straight  line 
into  a  curve,  until  they  fall  to  the  ground.  The  great- 
er the  force  with  which  they  are  projected,  the  greater 
is  the  distance  they  are  carried  before  they  fall.  If  we 
suppose  a  body  carried  several  miles  above  the  surface 
of  the  Earth,  jind  there  projected  in  an  horizontal  di- 
rection, with  so  great  a  velocity  that  it  moves  more  than 
semidiameter  of  the  Earth  in  the  line,  it  would  take  to 


54  On  Gravity.  Sec.  3 

fall  to  the  Earth  by  gravity,  in  that  case,  if  there  were 
no  resisting  medium,  the  body  would  not  fall  to  the  Earth 
at  all ;  but  continue  to  circulate  round  the  Earth,  keep- 
ing always  the  same  path,  and  returning  to  the  point 
from  whence  it  was  projected  with  the  same  velocity 
with  which  it  moved  at  first.  We  find  that  the  Moon 
therefore  must  be  acted  upon  by  two  powers,  one  of 
which  would  cause  her  to  move  in  a  right  line,  another 
bending  her  motion  from  that  line  into  a  curve.  This 
attractive  power  must  be  seated  in  the  Earth,  for  there 
is  no  other  body  within  the  Moon's  orbit  to  draw  her.* 
The  attractive  power  of  the  Earth  therefore  extends 
to  the  Moon,  and  in  combination  with  her  projectile 
force,  causes  her  to  move  round  the  Earth  in  the  same 
manner,  as  the  circulating  body  above  supposed. 

The  Moons  of  Jupiter,  Saturn  and  Herschel,  are  ob- 
served to  move  around  their  primary  planets  ;  there- 
fore there  is  an  attractive  power  in  these  planets,  op. 
erating  on  their  Satellites  in  the  same  manner,  as  the 
attraction  of  the  Earth  operates  on  the  Moon.  All  the 
planets  and  Comets  move  round  the  Sun,  and  respect 
it  as  their  centre  of  motion,  therefore  the  Sun  must 
be  endowed  with  an  attracting  power,  as  well  as  the 


*  If  the  Moon  revolves  in  her  orbit  in  consequence  of  an  attractive 
power  residing  in  the  Earth,  she  ought  to  be  attracted  as  much  from  the  tan- 
gent of  her  orbit  in  a  minute,  as  heavy  bodies  fall  at  the  Earth's  surface  in 
a  second  of  time.  It  is  accordingly  found  by  calculation,  that  the  Moon 
is  deflected  from  the  tangent  16,09  feet  in  a  minute,  which  is  the  very 
space  through  which  heavy  bodies  descend  in  a  second  of  time  at  the 
Earth's  surface. 


Sec.  3  On  Gravity.  55 

Earth  and  planets.  Consequently  all  the  bodies,  or 
matter  of  the  Solar  System  are  possessed  of  this  attrac- 
tive power,  and  also  all  matter  whatsoever. 

As  the  Sun  attracts  the  planets  with  their  Satellites;, 
and  the  Earth  the  Moon,  so  the  planets  and  Satellites 
re-attract  the  Sun,  and  the  Moon  the  Earth.  This  is 
also  confirmed  by  observation ;  for  the  Moon  raises 
tides  in  the  Ocean  ;  the  satellites  and  planets  disturb 
each  other's  motions.  Every  particle  of  matter  being 
possessed  of  an  attracting  power,  the  effect  of  the  whole 
must  be  in  proportion  to  the  quantity  of  matter  in  the 
body. 

Gravity  also,  like  all  other  virtues,  or  emanations,  ei- 
ther drawing  or  impelling  a  body  towards  a  centre,  de- 
creases as  the  square  of  the  distance  increases  ;  that 
is,  a  body  at  twice  the  distance,  attracts  another  with 
only  a  fourth  part  of  the  force  ;  at  four  times  the  dis- 
tance, with  a  sixteenth  part  of  the  force. 

By  considering  the  law  of  gravitation  which  takes 
place  throughout  the  Solar  System,  it  will  be  evident 
that  the  Earth  moves  round  the  Sun  in  a  year.  It  has 
been  stated  and  shown,  that  the  power  of  gravity  de- 
creases as  the  square  of  the  distance  increases,  and 
from  this  it  follows  with  mathematical  certainty,  that 
when  two  or  more  bodies  move  round  another  as  their 
centre  of  motion,  the  squares  of  the  time  of  their  peri- 
odical revolutions,  will  be  in  proportion  to  each  other, 
as  the  cubes  of  their  distances  from  the  central  body. — 
This  holds  precisely  with  regard  to  the  planets  round 


56  On  Gravity.  Sec.  3 

the  Sun,  and  the  satellites  round  their  primaries,  the 
relative  distances  of  which  are  well  known. 

All  Globes  which  turn  on  their  own  axis,  will  be  ob- 
late spheroids,  that  is,  their  surfaces  will  be  further 
from  their  centres  in  the  equatorial,  than  in  the  polar 
regions  ;  for  as  the  equatorial  parts  move  with  greater 
velocity,  they  will  recede  farthest  from  the  axis  of  mo 
tion,  and  enlarge  the  equatorial  diameter.  That  our 
Earth  is  really  of  this  figure,  is  demonstrable  from  the 
unequal  vibrations  of  a  pendulum,  and  the  unequal 
length  of  degrees  in  different  latitudes. 

Since  then  the  Earth  is  higher  at  the  equator  than  at 
the  poles,  the  Seas  naturally  would  run  towards  the 
polar  regions,  and  leave  the  equatorial  parts  dry,  if  the 
centrifugal  force  of  these  parts,  by  which  the  waters 
were  carried  thither,  did  not  keep  them  from  return- 
ing. Bodies  near  the  poles  are  heavier  than  these  near- 
er the  Earth's  centre,  where  the  whole  force  of  the 
Earth's  attraction  is  accumulated.  They  are  also 
heavier,  because  their  centrifugal  force  is  less  on  ac- 
count of  their  diurnal  motions  being  slower.  For  both 
these  reasons,  bodies  carried  from  the  poles  towards 
the  Equator,  gradually  lose  part  of  their  weight 

Experiments  prove  that  a  pendulum  which  vibrates 
seconds  near  the  poles,  vibrates  slower  near  the  Equa- 
tor, which  shows  that  it  is  lighter,  or  less  attracted 
there.  To  make  it  oscillate  in  the  same  time,  it  is 
found  necessary  to  diminish  its  length.  By  comparing 
the  different  lengths  of  pendulums  vibrating  seconds 
at  the  equator,  and  at  London  ;  it  is  found  that  a  pen- 


Sec.  3  Interrogations  for  Section  Third.  57 

dulum  must  be  2,542  lines  *   shorter  at  the  Equator 
than  at  the  poles. 


Interrogations  for  Section  Third. 

What  is  GRAVITY  ? 

Do  falling  bodies  strike  the  surface  of  the  Earth  at 
right  angles  1 

Do  falling  bodies  near  the  Earth,  always  direct  their 
course  to  its  centre  ? 

Where  is  the  centre  of  Gravity  situated  1 

When  bodies  are  projected  in  a  right  line,  what 
brings  them  to  the  Earth  ? 

If  there  were  no  attractive  power  at  the  centre  of 
the  Earth,  what  would  be  the  consequence  were  a  bo- 
dy so  projected,  and  not  meeting  any  resistance  from 
the  air? 

We  find  that  the  Moon  moves  round  the  Earth  in  an 
orbit  nearly  circular.  Why  is  it  so  1 

Where  is  that  attractive  power  situated  ? 

Have  the  other  planets  attractive  powers  also  * 

How  is  it  known  1 

*  A  line  is  l-12th  part  of  an  inch. 
O 


58  Interrogations  for  Section  Third.  Sec.  3 

Where  is  the  centre  of  attraction  of  the  Solar  Sys- 
tem placed  1 

How  is  it  known  1 

Do  the  planets  attract  the  Sun  as  well  as  the  Sun 
the  planets  ? 

Has  every  particle  of  matter  an  attractive  power  ? 

In  what  proportion  does  Gravity  increase  1 

How  far  is  the  Moon  deflected  by  Gravity  from  a 
tangent  in  one  minute  of  time  1 

How  far  does  a  failing  body  descend  in  one  second  1 

In  what  proportion  are  the  squares  of  the  times  of 
the  periodical  revolutions  of  all  the  planets  ? 

What  will  be  the  form  all  planets  which  revolve  on 
their  own  axis  1 

Why  will  they  be  of  that  form  ? 

How  is  it  ascertained  to  a  certainty,  that  our  Earth 
is  of  that  form  ? 

Why  are  bodies  near  the  poles  heavier  than  those  at 
the  Equator  1 

Why  is  a  pendulum  vibrating  seconds  shorter  at  the 
Equator  than  at  the  poles  ? 

What  is  the  length  of  a  pendulum  vibrating  seconds 
at  the  Equator  1 

Jlns.  39,2  inches. 


SECTION  FOURTH. 

PHENOMENA  OF  THE  HEAVENS,  AS  SEEN  FROM  DIFFERENT 
PARTS  OF  THE  EARTH. 

THE  magnitude  of  the  Earth  is  only  a  point  when  com- 
pared to  the  Heavens,  and  therefore  every  inhabitant 
upon  it,  let  him  be  in  any  place  on  its  surface,  sees  half 
of  the  Heavens.  The  inhabitant  on  the  North  Pole  of 
the  Earth,  constantly  sees  the  Northern  Hemisphere, 
and  having  the  North  Pole  of  the  Heavens  directly  over 
his  head,  his  horizon  coincides  with  the  celestial  Equa- 
tor. Therefore  all  the  Stars  in  the  Northern  Hemi- 
sphere, between  the  Equator  and  the  North  Pole,  ap- 
pear to  turn  round  parallel  to  the  horizon.  The  Equa- 
torial Stars  keep  in  the  horizon,  and  all  those  in  the 
Southern  Hemisphere  are  invisible.  The  like  phe- 
nomena are  seen  by  an  observer  at  the  South  Pole. — 
Hence,  under  either  pole,  only  one  half  of  the  Heavens 
is  seen ;  for  those  parts  which  are  once  visible  never 
set,  and  those  which  are  once  invisible  never  rise. 


60  Phenomena  of  the  Heavens,  fyc  /Sec.  4 

But  the  ecliptic  or  orbit,  which  the  Sun  appears  to  de- 
scribe once  a  year  by  the  annual  motion  of  the  Earth, 
has  the  half  constantly  above  the  horizon  of  the  north 
pole ;  and  the  other  half  always  below  it.  Therefore 
whilst  the  Sun  describes  the  northern  half  of  the  eclip- 
tic, he  neither  sets  to  the  north  pole,  nor  rises  to  the 
south ;  and  whilst  he  describes  the  southern  half,  he 
neither  sets  to  the  south  pole,  nor  rises  to  the  north.— 
The  same  observations  are  true  with  respect  to  the 
Moon,  with  this  difference  only,  that  as  the  Sun  de- 
scribes the  ecliptic  but  once  a  year,  he  is,  during  half 
that  time,  visible  to  each  pole  in  its  turn,  and  as  long 
invisible. 

But,  as  the  Moon  goes  round  the  ecliptic  in  27  days, 
8  hours,  she  is  only  visible  during  13  days  and  16  hours, 
and  as  long  invisible  to  each  pole  by  turns. 

All  the  planets  likewise  rise  &,  set  to  the  polar  regions 
because  their  orbits  are  cut  obliquely  in  halves  by 
the  horizon  of  the  poles.  When  the  Sun  arrives  at  the 
sign  Aries,  which  is  on  the  twentieth  of  March,  he  is 
just  rising  to  an  observer  on  the  north  pole,  and  set- 
ting to  another  on  the  south  pole.*  From  the  Equator, 
he  rises  higher  and  higher  in  every  apparent  diurnal 
revolution,  till  he  comes  to  the  highest  point  of  the 
ecliptic  on  the  21st  of  June,  and  then  he  is  at  his  great- 
est altitude,  which  is  23  degrees  and  28  minutes ; 


*  It  is  therefore  evident  when  the  Sun  is  on  the  Equator,  an  observer 
placed  at  each  pole,  sees  about  one  half  of  the  Sun  above  the  horizon,  and 
likewise  <xn  observer  at  the  Equator  discovers  both  poles  in  the  horizon. 


Sec.  4  Phenomena  of  the  Heavens,  fyc.  61 

equal  to  his  greatest  north  declination,  and  from  thence 
he  seems  gradually  to  descend  in  every  apparent  cir- 
cumvolution till  he  sets  at  the  sign  Libra,  on  the  23d 
September,  and  then  he  goes  to  exhibit  the  same  ap- 
pearances at  the  south  pole  for  the  other  half  of  the 
year.  Hence  the  Sun's  apparent  motion  round  the 
earth  is  not  in  parallel  circles,  but  in  spirals,  such  as 
may  be  represented  by  a  thread  wound  round  a  globe 
from  one  tropic  to  the  other.  If  the  observer  be*  any 
where  on  the  terrestrial  equator,  he  is  in  the  plane  of 
the  celestial  equator  or  under  the  equinox,  and  the  axis 
of  the  earth  is  coincident  with  the  plain  of  his  horizon 
extended  to  the  north  and  south  poles  of  the  Heavens. 
As  the  earth  performs  her  diurnal  revolution  on  her 
axis  from  west  to  east,  the  whole  Heavens  seem  to 
turn  round  the  contrary  way.  It  is  therefore  plain 
that  the  observer  at  the  equator  has  the  celestial  poles 
constantly  in  his  horizon,  and  that  his  horizon  cuts  the 
diurnal  paths  of  all  the  celestial  bodies  perpendicularly, 
and  in  halves.  Therefore,  the  Sun,  planets  and  stars 
rise  every  day,  and  ascend  perpendicularly  above  the 
horizon  for  six  hours,  and  passing  over  the  meridian, 
descend  in  the  same  manner,  for  the  six  hours  follow- 
ing, then  set  in  the  horizon  and  continue  12  hours  be- 
low it ;  consequently  the  days  and  nights  are  equally 
long  throughout  the  year.  Thus  we  find,  that  to  an 
observer  at  either  of  the  poles,  one  half  of  the  sky  is 
always  visible,  and  the  other  half  never  seen  ;  but  to  an 
observer  on  the  equator,  the  whole  sky  is  seen  every 
24  hours.  From  the  preceding  observations,  it  is  er- 


62  Phenomena  of  the  Heavens,  8fc.  Sec.  4 

ident,  that  as  the  Sun  advances  from  the  equator  to  the 
tropic  of  Cancer,  the  days  continually  lengthen/and 
the  nights  shorten  in  the  northern  hemisphere,  and  the 
contrary  in  the  southern  ;  and  when  the  Sun  descends 
from  the  equator  to  the  tropic  of  Capricorn,  the  days 
continually  lengthen  in  the  southern  hemisphere,  and 
the  nights  shorten ;  and  the  contrary  in  the  northern. 

The  earth's  orbit  being  elliptical,  and  the  Sun  con- 
stantly keeping  in  its  lower  focus,  which  is  one  million, 
three  hundred  and  seventy-seven  thousand  miles  from 
the  middle  point  of  the  longer  axis,  the  earth  comes 
twice  that  distance,  or  2,754,000  miles  nearer  the 
Sun*  at  one  time  of  the  year  than  at  another  ;  for  the 
Sun  appearing  under  a  larger  anglef  in  our  winter 
than  summer,  proves  that  the  earth  is  nearer  the  Sun 
in  winter  than  in  summer.  The  Sun  is  about  7  days 
longer  in  the  northern  hemisphere  than  in  the  southern 
in  every  year  ;  and  as  the  earth  approaches  near  to  the 
Sun,  its  motion  is  accelerated,  and  therefore  goes  over 
an  equal  space  in  less  time,  and  as  the  earth  recedes 
from  the  Sun,  its  motion  is  retarded  in  the  same  ratio 
that  it  was  accelerated  when  in  the  southern  hemi- 
sphere, and  consequently  requires  a  longer  time  to  pass 
over  an  equal  space. 


*  The  Sun  is  nearest  to  the  Earth  when  he  is  on  the  tropic  of  Capri- 
corn ;  farthest  from  it  when  he  is  on  the  tropic  of  Cancer. 

f  The  nearer  an  object  is  to  the  eye,  the  larger  it  appears,  and  under 
the  greater  angle  it  is  seen. 


Sec.  4  Phenomena  of  the  Heavens,  Sfc.  63 

But  here  a  question  actually  arises ;  why  have  we 
not  the  hottest  weather  when  the  earth  is  nearest  to  the 
Sun  1  In  answer,  it  must  be  observed,  that  the  eccen- 
tricity of  the  earth's  orbit  bears  no  greater  proportion 
to  the  earth's  mean  distance  from  the  Sun,  than  seven- 
teen bears  to  a  thousand,  and  therefore  this  small  differ- 
ence of  distance  cannot  occasion  any  great  difference 
of  heat  or  cold. 

But  the  principal  cause  of  this  difference  is,  that  in 
winter,  the  same  rays  fall  so  obliquely  upon  us,  that 
any  given  number  of  them  is  spread  over  a  much  great- 
er portion  of  the  earth's  surface  which  we  inhabit,  and 
therefore  each  point  must  then  have  less  rays  than  in 
summer.  Also  there  comes  a  greater  degree  of  cold 
in  the  long  winter  nights,  than  there  can  return  of  heat 
in  so  short  days,  and  on  both  these  accounts,  the  cold 
must  increase.  But  in  the  summer  season,  the  Sun's 
rays  fall  more  perpendicularly  upon  us,  aud  therefore 
come  with  greater  force,  and  in  greater  numbers  on 
the  same  place,  and  by  their  long  continuance,  a  much 
greater  degree  of  heat  is  imparted  by  day  than  can  fly 
off  by  night.  It  is  for  this  reason  that  we  have  a 
greater  degeee  of  heat  in  the  month  of  September,  than 
in  the  month  of  March  ;  the  Sun  being  on  the  equator 
in  both  these  months,  and  consequently  equally  distant 
from  the  earth.  Those  parts  which  are  once  heated, 
retain  the  heat  for  some  time,  which,  with  the  addi- 
tional heat  daily  imparted,  makes  it  continue  to  in- 
crease, though  the  Sun  declines  towards  the  south,  and 
this  is  the  reason  why  we  have  greater  heat  in  July 


64  Phenomena  of  the  Heavens,  fyc.  See.  4 

than  in  June.  Also,  we  know  that  the  weather  is  gen- 
erally warmer  at  2  o'clock  in  the  afternoon,  when  the 
Sun  has  gone  towards  the  west,  than  at  noon,  when 
he  is  on  the  meridian.  Likewise  those  places  which 
are  well  cooled,  require  time  to  be  heated  again,  for 
the  Sun's  rays  do  not  heat  even  the  surface  of  any  bo- 
dy till  they  have  been  sometime  upon  it,  and  therefore 
January  is  generally  colder  than  December  ;  although 
the  Sun  has  withdrawn  from  the  winter  tropic,  and 
begins  to  dart  his  beams  more  perpendicularly  upon 
us. 

It  was  formerly  the  opinion  of  Philosophers,  that  the 
Sun  was  an  immense  mass  of  flame,  and  consequently 
the  nearer  we  approached,  the  greater  must  be  the 
heat.  It  is  stated  that  the  heat  of  the  planet  Mercury 
is  seven  times  as  great  as  ours,  judging  from  his  near- 
ness to  the  Sun,and  likewise  that  the  cold  at  the  planets 
Jupiter,  vSaturn  and  Herschel,  must  be  extreme,  be- 
cause they  are  placed  at  so  great  a  distance  from  that 
luminary. 

It  is  well  known,  that  near  the  equator,  the  tops  of 
the  highest  mountains  are  covered  with  perpetual 
snow,  and  that  in  a  less  distance  than  three  miles  a- 
bove  the  surface  of  the  earth,  we  come  to  the  region 
of  perpetual  congelation,  where  neither  ice  nor  snow 
would  ever  melt,  although  nearer  the  Sun  than  in  the 
plane  below.  Therefore,  the  distance  from  the  Sun  is 
not  the  real  cause  of  heat  or  cold.  Dark  spots  have 
been  seen  upon  the  Sun's  disk,  from  which  it  is  gener- 
ally concluded  that  the  body  of  the  Sun  is  dark  and 


Sec.  4  Phenomena  of  the  Heavens,  fyc.  65 

opaque,  surrounded  by  a  luminous  atmosphere,  which 
darts  its  rays  with  immense  velocity,  and  by  some 
chemical  operation,  performed  in  their  passage  through 
the  atmosphere  with  which  this  earth  is  encircled, 
convey  to  us  the  sensation  of  heat.  The  solar  ob- 
servations of  Dr.  Wilson,  first  suggested  the  opinion, 
that  the  Sun  was  an  opaque  and  solid  body,  surrounded 
with  a  luminous  atmosphere,;"and  the  telescopes  of  Dr, 
Herschel  have  tended  still  farther  to  establish  this  opin- 
ion. The  latter  of  these  astronomers,  therefore  imagin- 
ed, that  the  functions  of  the  Sun,  as  the  source  of  light 
might  be  performed  by  the  agency  of  the  external  at- 
mosphere, while  the  solar  nucleus  was  reserved,  and 
fitted  for  the  reception  of  inhabitants.  That  the  Sun 
may  at  the  same  time  be  the  source  of  light  and  heat, 
and  yet  capable  of  supporting  animal  life  is  one  of  those 
conclusions,  which  we  are  fond  of  admitting  without 
hesitation,  and  to  cherish  writh  peculiar  complacency. 
The  mind  is  filled  with  admiration  of  the  wisdom  of 
that  Benign  Benefactor,  and  swrells  with  the  most  sub- 
lime emotions,  when  it  conceives,  that  apparently  the 
most  inaccessible  regions  of  creation  are  peopled  with 
animated  beings,  and,  that  while  the  Su»  is  the  foun- 
tain of  the  most  destructive  of  all  the  elements,  it  is  at 
the  same  time  Jhe  abode  of  life  and  plenty.  When  the 
invention  of  the  telescope  enabled  Astronomers  to  de- 
tect the  striking  resemblances  between  the  different 
planets  of  the  system,  it  was  natural  to  conclude,  that, 
as  they  were  composed  of  similar  materials,  as  they 


66  Phenomena  of  the  Heavens,  fyc.  Sec.  4 

revolved  around  the  same  centre,  and  were  enlighten- 
ed by  similar  Moons,  they  were  all  intended  by  their 
wise  Creator  to  be  the  region  in  which  he  chose  to 
dispense  the  blessings  of  existence  and  intelligence  to 
various  orders  of  animated  beings.  The  human  mind 
cheerfully  embraced  this  sublime  view  of  creation,  and 
guided  by  the  principle,  that  nothing  was  made  in  vain  ; 
man  extended  his  views  to  the  remote  corners  of  space, 
and  perceived  in  every  star  that  sparkles  in  the  sky, 
the  centre  of  a  magnificent  system  of  bodies,  teeming 
with  life  and  happiness,  and  displaying  fresh  instances 
of  the  power  and  beneficence  of  that  Being  who  rolled 
such  stupenduous  orbs  from  his  creating  hand. 

Having  thus  traversed  the  illimitable  regions  of 
space,  and  considering  every  w^orld  which  rolls  in  the 
immense  void  as  the  scene  on  which  the  Almighty  has 
exhibited  his  perfections,  the  mind,  unable  to  command 
a  wider  range,  rests  in  satisfaction  on  the  faithful  analo- 
gies which  it  has  pursued. 


Sec.  4          Interrogations  for  Section  Fourth.  67 


Interrogations  for  Section  Fourth. 

How  much  of  the  Heavens  does  a  spectator  see  pla- 
ced at  the  north  pole  of  the  earth  ? 

What  part  if  placed  at  the  south  pole  1 

What  part  if  seen  at  the  equator  ? 

When  placed  at  the  north  pole,  how  far  south  in  the 
Heavens  would  his  vision  extend  1 

If  at  the  south  pole,  how  far  north  would  his  vision 
extend  ? 

Does  any  part  of  the  ecliptic,  or  orbit,  which  the  Sun 
seems  to  describe  once  a  year,  appear  above  the  hori- 
zon of  the  north  pole  ? 

What  signs  of  the  ecliptic  appear  above  the  horizon 
of  the  north  pole '1 

jfe.--Aries,  Taurus,  Gemini,  Cancer,  Leo  and  Virgo; 
these  are  called  northern  sighs. 

What  signs  appear  above  the  horizon  of  the  south 
pole  ? 

Jlns. — Libra,  Scorpio,  Sagitarius,  Capricornus,  Aqua- 
rius, and  Pisces.     These  are  called  southern  signs. 

When  the  Sun  is  in  any  of  the  northern  signs,  does 
he  ever  set  at  the  north  pole,  or  rise  at  the  south  1 

When  in  the  southern  signs,  does  he  ever  set  to  the 
south  pole,  or  rise  to  the  north  1 

Is  it  the  same  with  the  Moon  ? 

What  is  the  length  of  the  longest  days  at  either  of  the 
poles  ? 


68  Interrogations  for  Section  Fourth.  See.  4 

What  the  length  of  the  longest  nights  ? 

In  what  month  of  the  year  is  the  Sun  on  the  tropic 
of  Cancer,  or  highest  point  of  his  orbit  ? 

In  what  months  on  the  tropic  of  Capricorn  ? 

In  what  months  on  the  equator  1 

When  the  Sun  is  in  the  equator,  can  he  be*  seen   at 
both  poles  ? 

Is  the  Sun's   apparent   motion  round  the   earth  in 
circles  ? 

Are  the  whole  Heavens  visible  every  24  hours  at  the 
equator  ? 

What  is  the  form  of  the  earth's  orbit.? 

How  many  miles  is  the  earth  nearer  the  Sun  at  one 
time  of  the  year  than  at  another  ? 

In  what  month  in  the  year  is  it  the  nearest  ? 

In  what  month  farthest  off? 

How    many  days  is  the  Sun  longer  in  the  northern 
hemisphere,  than  in  the  southern  in  every  year  ? 

Why  is  itlonger  north  of  the  equator  than  south  ? 

Why  have  we  not  the  warmest  weather  when  nearest 
the  Sun? 

What   was  the  opinion  of  former  Philosophers  con- 
cerning the  Sun  1 

What  tbe  opinion  of  Dr.  Hcrschel  ? 

How  far  above  the  surface  of  the  earth  at  the  equa- 
tor, is  the  region  of  perpetual  congelation  ? 

What  is   the  prevailing  opinion  concerning  the  rays 
of  the  Sun  producing  heat  ? 

Is  the  body  of  the  Sun  supposed  to  be  inhabited  ? 

With  what  is  its  dark  opaque  body  surrounded  1 
.  7 


SECTION  FIFTH. 

PHYSICAL  CAUSES  OF  THE  MOTIONS  OF  THE  PLANETS. 

FROM  the  uniform  projectile  motion  of  bodies  in 
straight  lines,  and  the  universal  power  of  attraction 
which  draws  them  off  from  these  lines,  the  curviiineal 
motions  of  all  the  planets  arise.  If  a  body  be  projected 
in  a  right  line  in  open  space,  and  meeting  with  no  re- 
sistance, it  would  continue  forever  to  move  with  the 
same  velocity,  and  in  the  same  direction.  But  when 
this  projectile  force  is  acted  upon  by  any  attracting  bo- 
dy, with  a  power  duly  adjusted,  and  perpendicular  to 
its  motion,  it  will  then  be  drawn  from  a  straight  line 
and  forced  to  revolve  round  the  centre  of  attraction  in 
a  circular  form.  But  when  the  projectile  force  first 
given  exceeds  the  attracting  force,  the  centrifugal  force, 
or  tendency  to  fly  off  in  a  tangent,  is  arrested  by  the  at- 
tractive power,  and  therefore  its  velocity  becomes  con- 
tinually more  and  more  impeded,  until  the  attractive 
power  has  acquired  a  greater  influence,  and  then  its 


70  Physical  causes  of  the  Motions  of  the  Planets.  Sec.  5 

motion  becomes  gradually  accelerated  with  a  tendency 
to  approach  nearer  to  its  point  of  attraction,  and  con- 
sequently the  moving  body  forms  an  elliptical  orbit. 

It  is  readily  perceived,  that,  in  two  points  in  the  or- 
bit, the  centripetal  and  centrifugal  forces  are  equal,  and 
that  when  its  motion  is  both  accelerated  and  retarded 
by  the  attractive  power,  it  must  of  necessity  pass  over 
equal  areas  in  the  same  space  of  time. 

As  the  planets  approach  nearer  the  sun,  and  recede 
farther  from  him  in  every  revolution,  there  may  be 
some  difficulty  in  conceiving  the  reason,  why  the  pow- 
er of  gravity  when  iConce  obtains  the  victory  over  the 
projectile  force,  does  not  bring  the  planets  continually 
nearer  the  sun  in  every  revolution,  till  they  fall  upon  and 
unite  with  him.  Or  why  the  projectile  force  when  it 
once  gets  the  better  of  gravity,  does  not  continue  to 
carry  the  planets  farther  from  the  sun,  till  it  removes 
them  quite  out  of  the  sphere  of  his  attraction  and  go  on 
in  straight  lines  forever  afterwards,  or  when  the  cen- 
tripetal and  centrifugal  forces  are  equal,  why  it  does  not 
commence  moving  off  in  the  form  of  a  perferct  circle m 
By  considering  the  effects  of  these  powers  acting  on 
each  other  according  to  the  preceding  description,  the 
difficulty  will  be  at  once  removed. 

A  double  projectile  force  will  always  balance  a  quad- 
ruple power  of  gravity,  and  when  a  planet  is  put  in 
motion  by  projectile  force,  whether  the  velocity  with 
which  it  moves  be  rapid  or  slow,  it  is  continually  resis- 
ted by  the  attraction  of  the  sun,  and  consequently 
moves,  slower  until  the  power  of  attraction  has 


Sec.  5  Physical  causes  of  the  Motion  of  the  Planets.  71 

gained  the  ascendency ;  its  motion  then  becomes  ac- 
celerated by  the  centripetal  power  acting  on  the  planet, 
until  its  velocity  becomes  equal  to  the  projectile  force 
with  which  it  was  first  put  in  motion.  Therefore  it 
must  continue  to  revolve  in  an  elliptical  orbit  as  before 
stated.  And  when  these  two  forces  are  equal  on  the 
body  in  motion,  they  never  act  at  right  angles,  but  in 
such  acute  angles,  that  the  planet  is  moving  with  such 
velocity,  that  the  ascendency  is  instantly  obtained.  In 
order  to  make  the  projectile  force  balance  the  gravita- 
ting power  so  exactly,  as  that  the  body  may  move  in  a 
circle,  the  projectile  velocity  of  the  body  must  be  such 
as  it  would  have  acquired  by  gravity  alone,  in  falling 
through  half  the  radius  of  the  circle. 

By  the  above  mentioned  law,  bodies  will  move  in  all 
kinds  of  elliptical  orbits  whether  long  or  short  ;  if  the 
spaces  in  which  they  move  in  the  longer  ellipses  have 
so  much  the  less  projectile  force  impressed  upon  them 
in  the  higher  parts  of  their  orbits,  and  their  velocities  in 
coming  down  towards  the  sun,  are  so  prodigiously  in- 
creased by  his  attraction,  that  their  centrifugal  forces 
in  the  lower  parts  of  their  orbits  are  so  great  as  to 
overcome  the  sun's  attraction  there,  and  cause  them  to 
ascend  again  towards  the  higher  parts  of  their  orbits; 
during  which,  the  sun's  attraction  acting  so  contrary  to 
the  motions  of  those  bodies,  causes  them  to  move  slow- 
er and  slower  until  the  projectile  forces  are  diminished, 
almost  to  nothing,  and  then  they  are  again  brought 
back  by  the  sun's  attraction  as  before. 

If  the  projectile  forces  of  all  the  planets  (and   like- 


72  Physical  causes  of  the  Motion  of  the  Planets.  Sec.  5 

wise  those  comets,  whose  mean  distances  from  the 
sun  have  been  ascertained,)  were  destroyed  at  their 
mean  distances  from  the  sun ;  their  gravities 
would  bring  them  down,  so  that  Mercury  would  fall 
to  the  sun  in  15  days  and  13  hours  ;  Venus  in  39  days 
and  17  hours  ;  the  Earth  or  Moon  in  64  days  and  10 
hours;  Mars  in  121  days;  Jupiter  in  290  days;  Sat- 
urn in  767  days ;  and  Herschel  in  5406  days ;  the 
nearest  comet  within  the  orbits  of  the  planets  in  13,- 
000  days ;  the  middlemost  in  23,000  days,  and  the 
outermost  in  66,000  days.  The  Moon  would  fall  to  the 
earth  in  4  days  and  20  hours,  Jupiters  first  moon 
would  fall  to  him  in  7  hours  ;  his  second  in  15  hours  ; 
his  third  in  30  hours ;  and  his  fourth  in  71  hours. — 
Saturns  first  moon  would  fall  to  him  in  8  hours,  his 
second  in  12  hours  ;  his  3d  in  19  hours  ;  his  4th  in  68 
hours  ;  his  5  in  336  hours.  A  stone  would  fall  to  the 
earth's  centre  if  there  were  an  hollow  passage  in  21 
minutes  and  9  seconds*. 

The  rapid  motions  of  the  moons  of  Jupiter  and  Sat- 
urn round  their  primaries,  demonstrate  that  these  two 
planets  have  stronger  attractive  power  than  the  earth  ; 
for  the  stronger  thatjone  body  attracts  another,  the 
greater  must  be  the  projectile  force,  and;, consequent- 
ly the  force  of  the  other  body  must  be  increased  to 


*  The  squares  of  the  times,  that  any  planet  would  fall  to  the  sun  are  as 
as  the  cubes  of  their  distances  :  or  multiply  the  time  of  a  whole  revolu- 
tion by  ,0176766  the  product  will  be  the  time  in  which  the  planet  would 
fall  to  the  sun. 


Sec.  5  Physical  causes  of  the  Motions  of  the  Planets.  73 

keep  it  from  falling  to  its  central  planet.  Jupiter's 
second  moon  is  124,000  miles  farther  from  Jupiter 
than  our  Moon  is  from  us ;  and  yet  this  second  Moon 
goes  almost  8  times  around  Jupiter,  whilst  our  Moon 
goes  once  round  the  Earth.  What  a  prodigious  at- 
tractive power  must  the  Sun  then  have,  to  draw  all 
the  planets  and  satellites  of  the  system  towards  him, 
and  what  an  amazing  power  must  it  have  acquired  to 
put  all  these  planets  and  moons  into  such  rapid  motions 
at  first.  Amazing  indeed  to  us,  because  impossible  to 
be  effected  by  the  united  strength  of  all  the  created  be- 
ings in  an  unlimited  number  of  worlds,  but  in  no  wise 
hard  for  the  Almighty,  whose  Planetarium  takes  in  the 
whole  Universe. 

The  Sun  and  planets  mutually  attract  each  other, 
the  power  by  which  they  are  thus  attracted,  is  called 
Gravity.  But  whether  this  power  be  mechanical  or 
not,  is  very  much  disputed.  Observation  proves  that 
the  planets  disturb  each  other's  motions  by  it,  and  that 
it  decreases  according  to  the  squares  of  the  distances 
of  the  Sun  and  planets  ;  as  great  light  which  is  known 
to  be  material,  likewise  does.  Hqnce,  Gravity  should 
seem  to  arise  from  the  agency  of  some  subtle  matter, 
pressing  towards  the  Sun  and  planets,  and  acting  like 
all  mechanical  causes,  by  contact.  But  when  we  con- 
sider that  the  degree  or  force  of  Gravity,  is  exactly  in 
proportion  to  the  quantities  of  matter  in  those  bodies, 
without  any  regard  to  their  magnitudes  or  quantities 
of  surface,  acting  as  freely  on  .their  internal  as  exter- 
nal parts,  it  appears  to  surpass  the  powers  of  mechan- 


74  Physical  causes  of  the  Motimis  of  the  Planets.  Sec.  5 

ism,  and  to  be  either  the  immediate  agency  of  the 
Deity,  or  affected  by  a  law  originally  established  and 
impressed  on  all  matter  by  him.  That  the  projectile 
force  was  at  first  given  by  the  Deity,  is  evident  ;  ince 
matter  can  never  put  itself  in  motion,  and  all  bodies 
may  be  moved  in  any  direction  whatever,  and  yet  the 
planets,  both  primary  and  secondary,  move  from  west 
to  east,  in  lines  nearly  coincident,  while  the  Comets 
move  in  all  directions,  and  in  planes  very  different  from 
each  other ;  these  motions  can  be  owing  to  no  me- 
chanical cause  or  necessity,  but  to  the  free  will  and 
power  of  an  intelligent  Being. 

Whatever  Gravity  be,  it  is  plain  that  it  acts  every 
moment  of  time ;  for  if  its  action  should  cease,  the 
projectile  force  would  instantly  carry  off  the  planets  in 
straight  lines  from  those  parts  of  their  orbits  where 
Gravity  left  them.  But  the  planets  being  once  put  in 
motion,  there  is  no  occasion  for  any  new  projectile 
force,  unless  they  meet  with  some  resistance  in  their 
orbits,  nor  for  any  mending  hand,  unless  they  disturb 
each  other  too  much  by  their  mutual  attraction. 

It  is  found  that  there  are  disturbances  among  the 
planets  in  their  motions,  arising  from  their  mutual  at- 
tractions, when  they  are  in  the  same  quarter  of  the 
Heavens,  and  the  best  modern  observers  find  that  our 
years  are  not  always  precisely  of  the  same  length. 

If  the  planets  did  not  mutually  attract  each  other, 
the  areas  described  by  them  would  be  exactly  propor- 
tional to  the  times  of  description.  But  observations 
prove  that  these  areas  are  not  in  such  exact  propor- 


Sec.  5  Physical  causes  of  the  Motion  of  the  Planets.  75 

tions,  and  are  most  varied  when  the  greatest  number 
of  planets  are  in  any  particular  quarter  of  the  Heav- 
ens. When  any  two  planets  are  in  conjunction,  their 
mutual  attractions  which  tend  to  bring  them  nearer  to 
each  other,  draws  the  inferior  one  a  little  nearer  to  him ; 
by  these  means,  the  figure  of  their  orbits  is  somewhat 
altered,  but  this  alteration  is  too  small  to  be  discovered 
in  several  ages.  By  the  most  simple  law,  the  diminu- 
ation  of  Gravity,  as  the  square  of  the  distance  increas- 
es ;  the  planets  are  not  only  retained  in  their  orbits, 
when  whirling  with  such  immense  velocity  around 
their  central  Sun  ;  but  an  eternal  stability  is  insured  to 
the  solar  system.  The  small  derangements  which  af- 
fect the  motions  of  the  Heavenly  bodies,  are  only  appa- 
rent to  the  eye  of  the  Astronomer,  and  even  these  after 
reaching  a  certain  limit,  gradually  diminish,  till  the  sys- 
tem regaining  its  balance,  returns  to  that  state  of  har- 
mony, and  order  which  has  preceded  the  commence- 
ment of  these  secular  irregularities.  Even  amidst  the 
changes  and  irregularities  of  the  system,  the  general 
harmony  is  always  apparent ;  and  those  partial  and 
temporary  derangements,  which  vulgar  minds  may 
seern  to  indicate  a  progressive  decay,  serve  only  to 
evince  the  stability  and  permanency  of  the  whole. 

In  contemplation  of  such  a  scene,  every  unperverted 
mind  must  be  struck  with  that  astonishing  wisdom 
which  framed  the  various  parts  of  the  Universe,  and 
bound  them  together  by  one  simple,  yet  infallible  law. 
In  no  part  of  creation,  from  the  smallest  insect  to  the 
highest  seraph,  has  the  Supreme  Architect  of  the  Uni- 


76  Interrogations  for  Section  Fifth.  Sec.  5 

verse  left  himself  without  a  witness  ;  but  it  is  surely  in 
the  Heavens  above,  that  the  Divine  attributes  are  most 
gloriously  displayed. 


Interrogations  for  Section  Fifth.  1 

From  what  source  do  the  circular  ^motions  of  the 
planets^arise  ? 

With  what  velocity  would  projected  bodies^continue 
to  move  if  they  met  with  no  resistance  ? 

What  isjneant  by  the  centrifugal  force  1 

What  is  meant  by  centripetal  force  ? 

Are  the  centripetal  and  centrifugal  forces  ever  equal 
while  the  planet  performs  its  revolution  round  the 
Sun? 

When  the  power  of  Gravity  exceeds  the  projectile 
force,  why  does  it  not  draw  the  planets  to  the  Sim  ? 

When  the  projectile,  or  centrifugal  force  exceeds  the 
attraction,  why  does  it  not  fly  oif,  and  never  return  ? 

When  these  forces  are  equal,  why  do  they  not  move 
in  perfect  circles  * 

What  will  a  double  projectile  force  balance  ? 

In  what  case  could  the  projectile  be  made  to  balance 
the  gravitating  power  in  such  manner  that  the  planets 
should  move  in  a  perfect  circle  ? 


Sec.  5  Interrogations  for  Section  Fifth.  77 

If  the  centrifugal  forces  should  at  once  be  destroyed, 
in  what  time  would  each  of  the  planets  fall  to  the  Sun  ? 

In  what  time  would  the  Moon  fall  to  the  earth  1 

What  rule  for  finding  the  time  in  which  would  they 
fall  to  the  Sun  ^ 

What  do  the  rapid  motions  of  the  Moons  of  Jupiter 
and  Saturn  demonstrate? 

Do  the  Sun  and  planets  continually  attract  each 
other  1 

Should  gravity  instantaneously  cease,  what  would 
be  the  consequence  ? 

Are  the  motions  of  the  planets  continually  the  same  ? 

Do  they  continue  to  move  exactly  in  the  same  path 
at  every  revolution  1 

By  what  simple  law  does  gravity  diminish  1 


SECTION  SIXTH. 
LIGHT  ./f  .7VXJ  JLIR. 


LIGHT  consists  of  exceedingly  small  particles  of 
matter,  issuing  from  a  luminous  body,  as  from  a  light- 
ed candle.  Such  particles  of  matter  constantly  flow 
in  every  direction.  By  Dr.  Neiwentyt's  computation, 
148,660,000,000,000,000,000,000,000,000,000,000,000, 
000,000  particles  of  light  in  one  second  of  time  flows 
from  a  candle,  which  number  contains  at  least  6,337,- 
245,000,000  times  the  number  of  grains  of  sand  in  the 
whole  earth;  supposing  100  grains  of  sand  to  be  equal 
in  length  to  an  inch,  and  consequently  every  cubic 
inch  of  the  earth  to  contain  one  million  of  such  grains. 
These  amazingly  small  particles,  by  striking  upon  our 
eyes,  excite  in  our  minds  the  idea  of  light,  and  if  they 
were  as  large  as  the  smallest  particle  of  matter  discern- 
ible by  our  best  microscopes,  instead  of  being  servicea- 
ble to  us,  they  would  soon  deprive  us  of  sight  by  the 


Sec.  6  On  Light  and  Heat.  79 

force  arising  from  their  immense  velocity,  which  is 
computed  at  nearly  two  hundred  thousand  miles  in  one 
second.* 

When  these  small  particles  flowing  from  a  candle, 
fall  upon  bodies,  and  are  thereby  reflected  to  our  eyes, 
they  excite  in  us  the  idea  of  that  body,  by  forming  its 
image  on  the  retina,  f  Since  bodies  are  visible  on  all 
sides,  light  must  be  reflected  from  them  in  all  direc- 
tions. A  ray  of  light  is  a  continued  stream  of  these 
particles,  flowing  from  any  visible  body  in  a  straight 
line.  That  the  rays  move  in  straight,  and  not  in  crook- 
ed lines,  (unless  they  be  refracted,)  is  evident  from  bo- 
dies not  being  visible  if  we  endeavor  to  look  at  them 
through  the  bore  of  a  bended  pipe ;  and  from  their 
ceasing  to  be  seen  by  the  interposition  of  other  bodies, 
as  the  fixed  stars,  by  the  interposition  of  the  Moon 
and  planets  and  the  Sun  wholly,  or  in  part,  by  the  in- 
ter position  of  the  Moon,  Mercury  or  Venus. 

There  is  no  physical  point, (says  Melville,)  in  the  vis 
ible  horizon  which  does  not  send  rays  to  every  other 
point;  no  star  in  the  Heavens,  which  does  not  send 
light  to  every  other  star.  The  whole  horizon  is  filled 
with  rays  from  every  point  in  it ;  and  the  whole  visible 
Universe  with  a  sphere  of  rays  from  every  star.  In 
short,  for  any  thing  we  know,  there  are  rays  of  light 
joining  every  two  physical  points  in  the  Universe,  and 


*  Light  passes  from  the  Sun  to  the   Earth  in  8  minutes  and  7  seconds, 
which  is  195,072  miles  in  one  second  of  time. 

f  A  fine  net  work  membrane,  in  the  bottom  of  the  eye. 


80  On  Light  and  Heat.  Sec.  6 

that  in  contrary  directions,  except  when  opaque  bodies 
intervene.  A  ray  of  light  coming  from  any  of  the  fixed 
stars  to  the  human  eye,  has  to  pass  in  every  part  of  the 
intermediate  space  between  the  point  from  which  it 
has  been  projected,  and  our  solar  system,  through  rays' 
of  light  flowing  in  all  directions  from  every  fixed  star  in 
the  Universe,  and  in  reaching  this  earth  ;  it  has  passed 
across  the  whole  ocean  of  the  solar  light,  and  that  which 
is  emitted  from  the  planets,  satellites  and  comets.  Yet 
in  this  course,  its  progress  has  not  been  intercepted. 

The  densities  and  quantities  of  light,  received  upon 
any  given  plane,  are  diminished  in  the  same  proportion, 
as  the  squares  of  the  distances  of  that  planet  from  the 
luminous  body  are  increased  ;  and  on  the  contrary,  are 
increased  in  the  same  proportion  as  these  squares  are 
diminished. 

When  a  telescope  magnifies  the  disk  of  the  Moon, 
and  planets,  they  appear  more  dim  than  to  the  bare 
eye  ;  because  the  telescope  cannot  increase  the  quanti- 
ty of  light  in  the  same  proportion  that  it  can  magnify 
the  surface,  and  by  spreading  the  same  quantity  of  light 
over  a  given  surface,  it  appears  more  dim,  than  when 
beheld  with  the  naked  eye. 

When  a  ray  of  light  passes  out  of  one  medium  into 
another,  it  is  refracted,  or  turned  out  of  its  course  more 
or  less,  as  it  falls  more,  or  less  obliquely  on  the  refrac- 
ting surface  which  divides  the  two  mediums. 

This  may  be  proved  by  several  experiments.  In  a 
basin,  place  a  piece  of  money,  or  any  metalic  sub- 
stance, and  then  retire  from  it  till  the  edge  of  the  basin 


Sec.  6  On  Light  and  Heat.  81 

hides  the  money  from  your  view,  then  keeping  ycur 
head  steady,  let  another  pour  water  gently  into  the  ba- 
sin, and  as  the  basin  fills  with  the  water,  more  and  more 
uf  (he  substance  of  the  bottom  will  come  in  sight,  and 
when  the  basin  is  filled,  the  substance  at  the  bottom 
will  be  full  in  view,  and  appear  as  if  it  was  lifted  up  ; 
for  the  ray  which  was  straight  while  the  basin  was  emp- 
ty, is  now  bent  at  the  surface  of  the  water,  and  turned 
out  of  its  natural  course  into  an  angular  direction,  and 
the  more  dense  the  medium  is,  the  more  light  is  reflect- 
ed in  passing  through  it. 

The  earth  is  surrounded  by  a  thin  fluid  mass  of  mat- 
ter, called  the  AIR,  or  ATMOSPHERE,  which  gravitates 
to  the  earth,  revolves  with  it  in  its  diurnal  motion,  and 
goes  with  it  round  the  Sun  every  year.  This  fluid  is 
of  an  elastic  and  springy  nature,  and  that  part  next  the 
earth  being  compressed  by  the  weight  of  all  the  air 
above  it,  is  pressed  close  together,  and  therefore  is  the 
most  dense  at  the  surface  of  the  earth,  and  gradually 
rarer  the  higher  you  ascend. 

It  is  well  known,  that  the  air  near  the  surface  of  our 
earth  possesses  a  space  about  nine  hundred  times  great- 
er than  water  of  the  same  weight,  and  therefore  a  cyl- 
indric  column  of  air  nine  hundred  feet  high,  is  of  equal 
weight  with  a  cylinder  of  water  of  the  same  diameter 
one  foot  high.  But  a  cylinder  of  air  reaching  to  the 
top  of  the  atmosphere,  (45  miles,)  is  of  equal  weight 
with  a  cylinder  of  water  about  33  feet  high,  and  there- 
fore, if  from  the  whole  cylinder  of  air,  the  lower  part 

of  nine  hundred  feet  high,  is  taken  away,  the  upper  part 

j 


82  On  Light  and  Heat.  Sec.  6 

remaining,  will  be  of  equal  weight  with  a  cylinder  of 
water  32  feet  high.  Therefore,  at  the  height  of  nine 
hundred  feet,  the  weight  of  the  incumbent  air  is  less, 
and  consequently  the  rarity  of  the  compressed  air  is 
greater,  than  near  the  earth's  surface  in  the  ratio  of  33 
to  32. 

The  weight  of  the  air  at  the  earth's  surface,  is  found, 
by  experiments  made  with  the  air  pump,  and  also  by 
the  quantity  of  mercury  that  the  atmosphere' balances  in 
the  barometer,  in  which/ at  a  mean  state,  the  mercury 
stands  29  and  J  inches  high.  And  if  the  tube  were  a 
square  inch  at  the  base,  and  of  equal  size  to  the 
top,  it  would,  at  that  height,  contain  29  and  4  cubic  in- 
ches of  mercury  ;  which  is  just  fifteen  pounds,  and  con- 
sequently every  square  inch  of  the  surface  of  the  earth, 
sustains  a  weight  of  15  pounds  ;  every  square  foot  2,- 
160  pounds,  at  this  ratio,  and  when  the  mercury  is  at 
that  height  in  the  barometer,  every  common  sized  man 
sustains  a  weight  of  32,400  pounds,  (the  area  of  the  sur- 
face of  his  body  being  about  15  square  feet)  of  air  all 
round;  for  fluids  press  equally  up  and  down,  and  on  all 
sides.  But  because  this  enormous  weight  is  equal  on 
all  sides,  and  counterbalanced  by  the  spring  of  the  air 
diffused  through  all  parts  of  our  bodies,  it  is  not  in  the 
least  degree  felt  by  us. 

The  state  of  the  air  is  such  many  times,  that  we  feel 
ourselves  languid  and  dull,  which  is  generally  thought 
to  be  occasioned  by  the  air's  being  foggy  and  heavy  a- 
bout  us.  But  at  such  times,  the  air  is  too  light.  The 
truth  of  this  assertion,  is  known  by  the  sinking  of  (be 


Sec.  6  On  Light  and  Heat.  83 

mercury  in  the  barometer,  and  at  these  times,  it  is  gen- 
erally found  that  the  air  has  not  sufficient  strength  to 
bear  up  the  vapors  which  compose  the  clouds ;  for 
when  it  is  otherwise  the  clouds  ascend  high,  and,  the 
air  is  more  elastic  and  weighty  about  us,  and  by  these 
means,  it  balances  the  internal  spring  of  the  air  within 
us,  braces  the  nerves  and  blood  vessels,  and  makes  us 
brisk  and  lively. 

It  is  entirely  owing  to  the  state  of  the  atmosphere,that 
the  Heavens  appear  bright  even  in  the  day  time.  For, 
without  an  atmosphere,  only  that  part  of  the  Heavens 
would  shine  in  which  the  Sun  was  placed,  and  if  we 
eould  live  without  air,  and  should  turn  our  backs  to- 
wards the  Sun,  the  whole  Heavens  would  appear  as 
dark  as  in  the  night ;  and  the  stars  would  be  seen  as 
clearly  as  in  the  nocturnal  sky.  In  this  case,  we  should 
have  no  twilight,  but  a  sudden  transition  from  the 
brightest  sunshine  to  the  darkness  of  night,  immediate- 
ly after  Sun-set,  and  from  the  blackest  darkness  to  the 
brightest  Sun-shine  at  Sun-rising. 

But,  by  means  of  the  atmosphere,  we  enjoy  the  Sun's 
light  reflected  from  the  aerial  particles  for  some  time 
before  he  rises,  and  after  he  sets.  When  the  earth  by 
its  rotation,  has  withdrawn  our  sight  from  the  Sun,  the 
atmosphere,  (being  still  higher  than  we,)  has  the  Sun's 
light  imparted  to  it  which  gradually  decreases  until*  he 
has  descended  18  degrees  below  the  horizon,  and  then 
all  that  part  of  the  atmosphere  which  is  above  us  is  dark. 
From  the  length  of  twilight,  Dr.  Heill  has  calculated 
the  height  of  the  atmosphere  (so  far  as  it  is  sufficiently 


84  On  Light  and  Heat.  Sec.  6 

dense  to  reflect  any  light,)  to  be  about  forty-four  miles 
high.  But  it  seldom  is  sufficiently  dense  at  two  miles 
height  to  bear  up  the  clouds, 

The  atmosphere  refracts  the  rays  so  as  to  bring  him 
in  sight  every  clear  day  before  he  rises  in  the  horizon, 
and  to  keep  him  in  view  for  some  minutes  after  he  is 
really  set  below  it.  For,  at  some  times  of  the  year,  we 
see  the  Sun  ten  minutes  longer  above  the  horizon  than 
he  would  be,  if  there  were  no  refractions,  and  about  six 
minutes  every  day.  at  a  mean  rate. 


Sec.  6  Interrogations  for  Section  Sixth.  85 


Interrogations  for  section  Sixth* 

Of  what  docs  LIGHT  consist  1 

What  would  be  the  consequence,  if  the  particles  of 
light  were  sufficiently  large  to  be  discovered  by  our  best 
microscopes. 

What  is  meant  by  the  retina  ? 

Is  light  reflected  in  all  directions  ? 

What  is  a  ray  of  light  1 

How  is  it  ascertained  that  rays  of  light  move  in  di- 
rect lines  ? 

Is  light  reflected  from  the  Moon  to  our  Earth  ? 

Is  light  sent  from  one  star  to  another  ? 

In  what  time  does  light  pass  from  the  Sun  to  the 
Earth  1 

How  many  miles  per  second  is  its  velocity  ? 

Are  the  rays  of  light  passing  from  any  luminous  body, 
interrupted  by  those  from  any  other  ? 

Are  the  quantities  of  light  received  upon  any  given 
plane,  diminished  by  being  removed  at  a  greater  distance 
from  that  plane  ? 

In  what  proportion  are  they  diminished  ? 

What  appearance  have  the  moons  or  planets,  when 
their  disks  are  magnified  by  the  aid  of  a  telescope  ? 

When  a  ray  of  light  passes  out  of  one  medium  into 
another,  is  it  turned  out  of  its  former  course  ? 


86  Interrogations  for  Section  Fourth.  Sec.  4 

If  it  does,  how  can  it  be  proved  ? 

With  what  is  the  earth  surrounded  ? 

What  is  the  nature  of  this  fluid  1 

Is  it  capable  of  being  compressed  1 

Is  it  more  dense  at  the  surface  of  the  earth,  than  some 
distance  above? 

How  much  heavier  is  water  than  atmospheric  air  ? 

What  is  the  mean  height  of  the  atmosphere  1 

By  what  means  is  the  weight  of  air  at  the  surface  of  the 
earth,  found  ? 

What  is  the  weight  of  air  on  every  square  inch  ? 

How  many  pounds  on  a  square  foot  ? 

How  many  on  the  surface  of  the  body  of  a  common 
sized  man  1 

In  what  kind  of  weather  is  the  air  lightest  ? 

How  is  it  proved  ? 

In  what  kind  the  heaviest  ? 

At  what  height  may  the  clouds  be  borne  by  the  density 
of  the  atmosphere  ? 

How  is  air  rarified  ? 

By  what  is  the  Sun  discovered,  when  he  is  in  reality 
below  the  horizon  ? 


SECTION  SEVENTH. 

TO  FIND  THE  DIAMETER  OF  THE  EARTH,  AND  THE  DISTAN- 
CES OF  THE  SUN  AND  MOON,  AND  THE  DISTANCES 
OF  ALL  THE  PLANETS  FROM  THE  SUN. 

IT  has  been  observed  that  a  person  at  Sea,  and  ex- 
actly under  the  equator,  discovers  both  the  north  and 
south  polar  stars,  just  rising  in  the  horizon.  There- 
fore as  you  advance  towards  either  pole,  the  star  will 
appear  to  rise  higher  in  the  horizon,  and  if  you  advance 
ten  degrees  from  the  equator  towards  the  north  pole, 
the  polar  star  will  there  be  ten  degrees  above  the  ho- 
rizon, and  consequently,  if  the  angle  of  the  elevation 
of  that  star  be  taken  at  any  place,  the  number  of  de- 
grees of  its  elevation,  will  be  equal  to  the  north  lati- 
tude of  the  place  where  the  elevation  was  taken.  It 
has  been  ascertained  by  actual  measurement,  that  one 
degree  of  the  surface  of  the  earth  contains  69  and  J 

NOTE — Persons  unacquainted  withTrigonometry,  may  pass  over  this 
Section,  as  they  will  not  be  capable  of  forming  correct  ideas  of  the  meth- 
ods of  finding  the  accurate  distances,  and  therefore  must  take  the  Philoso- 
pher's word. 


88       To  find  the  Distances  of  the  Planets,  8?c.       Sec.  1 

miles  nearly.  Then,  as  one  degree  is  to  69  and  A 
miles,  so  is  360  degrees  to  the  circumference  of  the 
earth.  The  diameter  can  then  be  found  by  the  follow- 
ing proportion  ;  as  355  is  to  the  circumference,  so  is 
113  to  the  diameter. 

Let  a  large  graduated  instrument  having  a  more 
able  index,  with  sight  holes,  be  prepared,  in  such  man- 
ner, that  its  plane  surface  may  be  parallel  to  the  plane 
of  the  equator,  and  its  edge  in  the  meridian,  so  that 
when  the  Moon  is  in  the  equinox,  and  on  the  meridian, 
she  may  be  seen  through  the  sight  holes  when  the 
edge  of  the  moveable  index  cuts  the  beginning  of  the 
divisions  on  the  graduated  limb,  and  when  she  is*  so 
seen,  let  the  precise  time  be  noted.  As  the  Moon  re- 
volves about  the  earth  from  any  meridian  to  the  same 
again  in  24  hours  and  48  minutes,  she  will  go  a  fourth 
part  round  it  in  a  fourth  part  of  that  time ;  namely  six 
hours  and  12  minutes  as  seen  from  the  earth's  centre 
or  pole  :  But,  as  seen  from  the  observer's  place  on  the 
earth's  surface,  the  Moon  will  seem  to  have  gone  a 
quarter  round  the  earth,  when  she  comes  to  the  sensi- 
ble horizon  ;  for  the  index  through  the  sights  of  which 
she  is  then  viewed,  will  be  90  degrees  from  where  it 
was  when  she  was  first  seen.  Let  the  exact  moment 
when  she  is  in,  or  near  the  sensible  horizon  be  careful- 
ly noteil,*  that  it  may  be  known  in  what  time  she  has 


*  Here  proper  allowances  must  be  made,  (for  the  refraction  being 
about  33  minutes  of  a  degree  in  the  horizon,)  will  cause  the  Moon's  centre 
to  appear  33  minutes  above  the  horizon,  when  her  centre  is  really  in  it. 


Sec.  7       To  find  the  Distances  of  the  Planets,  fyc.       89 

gone  through  90  degrees,  as  seen  by  the  observer  on 
the  surface  ;  subtract  this  time,  from  the  aforesaid  six 
hours  and  12  minutes,  and  the  remainder  is  the  time 
that  she  is  moving  in  her  orbit,  from  a  tangent,  touch- 
ing the  earth's  surface,  to  a  parallel  line  drawn  from 
the  earth's  centre  ;  which  affords  an  easy  matter  of 
finding  the  Moon's  horizontal  parallax,  which  is  equal 
to  an  angle  made  between  the  last  mentioned  line,  and 
another  drawn  from  the  observer  to  the  centre  of  the 
Moon,  as  seen  from  the  centre  of  the  earth  or  poles. — 
Then,  as  the  afore  mentioned  remainder  is  to  90  de- 
grees, so  is  6  hcurs  12  minutes,  to  the  number  of  de- 
grees which  measures  the  arc ;  subtract  99  de- 
grees from  this  arc,  and  the  remainder  is  the  angle  un- 
der which  the  earth's  semi-diameter  is  seen  from  the 
Moon.  Since  all  the  angles  of  a  right  angled  triangle 
are  equal  to  two  right  angles,  or  180  degrees,  and  the 
sides  of  a  plain  triangle  are  always  proportionate  to  the 
sines  of  their  opposite  angles ;  say,  (by  Trigonometry,) 
as  the  sine  of  the  angle  at  the  Moon,  is  to  the  earth's 
semi-diameter,  so  is  radius,  (or  sine  of  90  degrees,)  to 
its  opposite  side  ;  which  is  the  distance  from  the  obser- 
ver to  the  Moon,  or  subtract  the  angle  at  the  Moon  from 
90  degrees,  then  say  as  the  angle  at  the  Moon,  is  to 
the  earth's  semi-diameter,  so  is  this  remainder,  to  the 
distance  from  the  centre  of  the  earth  to  the  Moon ; 
which  comes  out  at  a  mean  rate  240  thousand  miles. 
The  Sun's  distance  from  the  earth  might  be  found  in 
the  same  manner,  if  his  horizontal  parallax  were  not 
so  small  as  to  be  hardly  perceptible,  being  8,63 


90       Tofnd  the  Distances  of  the  Planets,  $c.       Sec.  7 

seconds,*  while  the  horizontal  parallax  of  the  Moon  is 
57  minutes  and  18  seconds.  Therefore,  to  find  the  dis- 
tance to  the  Sun,  say  bj  single  proportion,  as  the  Sun's 
horizontal  parallax,  (8,63  seconds,)  is  to  the  distance 
that  the  Moon  is  from  the  earth,  (240,000  miles,)  so  is 
the  Moon's  horizontal  parallax,  (57  minutes  and  18 
seconds,)  to  the  distance  of  the  Sun  from  the  earth, 
which  gives  in  round  numbers,  95  millions  of  miles. 

The  Sun  and  Moon  appear  nearly  of  the  same  size 
as  viewed  from  the  earth,  and  every  person  who  un- 
derstands Trigonometry,  knows  how  their  true  mag- 
nitudes may  be  ascertained  from  the  apparent,  when 
their  true  distances  are  known. 

Spheres  are  to  each  other,  as  the  cubes  of  their  di- 
ameters. Whence,  if  the  Sun  be  95  millions  of  miles 
from  the  earth,  to  appear  of  equal  size  with  the  Moon, 
whcss  distance  is  only  240  thousand  ;  he  must  in  solid 
bulk,  be  62  millions  of  times  larger  than  the  Mccn. 

The  horizontal  parallaxes  are  best  observed  at  the 
equator ;  Because  the  heat  is  so  nearly  equal  every 
day,  that  the  refractions  are  almost  constantly  the  same, 
and  likewise,  because  the  parallactic  angle  is  greater 
there  ;  the  distance  from  thence  to  the  earth's  axis, 
being  greater  than  upon  any  parallel  of  latitude. 

The  earth's  distance  from  the  Sun  being  determin- 
ed, the  distances  of  all  the  other  planets  are  easily 
found  by  the  following  analogy  ;  their  pericdical  rev- 


*  Ascertained  from  the  transits  •!*  Venus  across  the  Sun's  disk,  in  ;te 
years  1761  and  1769. 


Sec.  7       Tjfind  the  Distances  of  the  Planets,  fyc.       91 

olutions  around  him,  being  obtained  by  observation. — 
As  the  square  of  the  earth's  period  round  the  Sun,is  to 
the  cube  of  its  distance  from  that  luminary,  so  is  the 
square  of  the  period  of  any  other  planet,  to  the  cube  of 
its  distance,  in  such  parts  or  measures,  as  the  earth's 
distance  was  taken.  This  proportion  gives  the  relative 
mean  distances  of  the  planets  from  the  Sun  to  the 
greatest  degree  of  exactness. 

The  earth's  axis  produced  to  the  stars,  and  being 
carried  parallel  to  itself  during  the  earth's  annual  rev- 
oluticn,  describes  a  circle  in  the  sphere  of  the  fixed 
stars,  equal  to  the  earth's  orbit.  This  orbit,  though 
very  large,  would  seem  to  be  no  larger  than  a  point, 
if  it  were  viewed  from  the  stars,  and  consequently,  the 
circle  described  in  the  sphere  of  the  stars,  by  the  axis 
of  the  earth,  produced,  if  viewed  from  the  earth, 
must  appear  as  a  point ;  its  diamater  appears  too  little 
to  be  measured  by  observation. 

Dr.  Bradley  has  assured  us,  that  if  it  had  amounted 
•  to  a  single  second,  or  two  at  most,  he  should  have  per- 
ceived it  in  the  great  number  of  observations  he  has* 
made ;  especially  upon  Draconis,  (a  star  of  the  third 
magnitude,)  and  that  it  seemed  to  him  very  probable 
that  the  annual  parallax  of  this  star,  is  not  so  great  as 
a  single  second,  and  consequently  that  it  is  more  than 
four  hundred  thousand  times  further  from  us  than  the 
Sun.  If  we  suppose  that  the  parallax  of  the  nearest 
fixed  star  is  one  second,  and  that  the  mean  distance  of 
the  earth  from  the  Sun,  is  95  millions  of  miles,  we 
shall  have  a  right  angled  triangle  whose  vertical  angle 
is  one  second,  and  whcwe  ba«e  is  95  millions  of  miles  to 


92       To  find  the  Distances  of  the  Planets,  fyc.       Sec.  7 

find  its  side,  or  distance  of  the  star  ;  which  would  ex- 
ceed 20  billions  of  miles,  a  distance  through  which  light 
although  travelling  at  the  rate  of  two  hundred  thousand 
miles  in  a  second,  could  not  pass  in  three  years.  If  the 
brightest  star  in  the  Heavens  is  placed  at  such  an  im 
mense  distance  from  our  system,  what  an  immeasure- 
able  interval  must  lie  between  us  and  those  minute 
stars,  whose  light  is  scarcely  visible  by  the  aid  of  the 
most  powerful  telescopes.  Many  of  them  are,  per- 
haps so  remote,  that  the  first  beam  of  light  which  they 
sent  forth  at  their  creation,  has  not  yet  arrived  within 
the  limits  of  our  system.  While  other  stars  which  have 
disappeared,  or  have  been  destroyed  for  many  centu- 
ries, will  continue  to  shine  in  the  Heavens  till  the  last 
ray  which  they  emitted,  has  reached  the  earth  which 
we  inhabit. 

The  mean  distances  of  the  planets  from  the  Sun,  and 
their  apparent  diameters  as  seen  from  that  luminary, 
being  found,  the  diameters  of  all  the  planets  can  be  as- 
certained by  Trigonometry  ;  thus — Subtract  the  angle 
or  apparent  diameter  of  the  planet  as  seen  from  the  sun, 
from  180  degrees,  and  half  the  remainder  will  be  the  an 
gle  at  the  disk  of  the  planet;  then  as  the  sine  of  the  angle 
at  the  disk  is  to  the  distance  of  the  planet  from  the  Sun, 
so  is  half  the  angle  at  the  Sun  to  the  semi-diameter  of 
the  planet. 

The  small  apparent  motion  of  the  stars  discovered 
by  that  great  Astronomer,  (Dr.  Bradley,),  he  found 
to  be  owing  to  the  aberation  of  their  light,  which  can 
result  from  no  known  cause,  except  that  of  the  earth'* 


Sec.  1  Interrogations  for  Section  Seventh.  93 

annual  motion,  as  it  agrees  so  exactly  therewith,  it 
proves  beyond  dispute  that  the  earth  has  such  a  mo- 
tion ;  for  this  aberation  completes  all  its  various  phe- 
nomena every  year,  and  proves  that  the  velocity  of 
star-light  is  such  as  carries  it  through  a  space  equal  to 
the  Sun's  distance  from  us,  in  eight  minutes  and  seven 
seconds  of  time.  Hence  the  velocity  of  light  is  about 
10,313  times  as  great  as  the  earth's  velocity  in  its  or- 
bit,^and  consequently  nearly  two  hundred  thousand 
miles  in  one  second  of  time. 


Interrogations  for  Section  Seventh. 

From  what  place  can  the  North  and  South  Poles 
both  be  seen  1 

Where  will  they  appear  1 

What  is  an  angle  of  elevation  ? 

Suppose  the  north  polar  star  is  elevated  43  degrees 
above  the  horizon,  what  is  your  degree  of  latitude  ? 

How  many  miles  constitute  a  degree  on  the  surface 
of  the  earth  1 

How  is  that  known  ? 

How  many  degrees  in  a  circle  1 

How  is  the  circumference  of  the  earth  found  7 

How  the  diameter  ? 


94  Interrogations  for  Section  Seventh.  Sec.  7 

In  what  time  doas  the  Moon  revolve  about  the  earth 
from  any  meridian  to  the  same  again  ? 

\\  hat  is  meant  by  the  Moon's  horizontal  parallax  ? 

How. is  the  distance  to  the  Moon  found  ? 

How  is  the  distance  to  the  Sun  found  ? 

\\  hat  proportions  do  solid  bodies  bear  to  each  other  ? 

When  the  distance  from  the  earth  to  the  Sun  is 
found,  how  is  the  distance  from  the  Sun  to  the  other 
planets  ascertained  ? 

By  what  method  are  the  periodical  revolutions  of 
the  planets  ascertained  ? 

How  are  the  periods  ascertained  of  their  revolutions 
on  their  own  axis  1 

Why  cannot  the  exact  distance  to  the  fixed  stars  be 

•f 

found  1 

How  are  the  diameters  of  the  planets  ascertained  ? 
How  the  diameter  of  the  Sun  ? 


SECTION  EIGHTH. 

OF  THE  EQUATION  OF  ^IME,  AND  PRECESSION  OF  THE 
EQUINOXES. 

THE  Stars  appear  to  go  around  the  earth  in  23  hours 
56  minutes  and  4  second?,  and  the  Sun  in  24  hours. 
So  that  the  stars  gain  3  minutes  and  56  seconds  upon 
the  Sun  every  day  ;  which  amounts  to  one  diurnal  rev- 
olution in  a  year,  or  365  days,  as  measured  by  the 
returns  of  the  Sun  to  the  meridian;  there  are  366  days 
as  measured  by  the  stars  returning  to  it.  The  former 
are  called  solar  days ;  the  latter  sydereal. 

The  earth's  motion  on  its  axis  being  perfectly  uni- 
form, and  equal  at  all  times  of  the  year  ;  the  sydereal 
days  are  always  precisely  of  an  equal  length,  and  so 
would  the  solar  days  be  if  the  earth's  orbit  wrere  a  per- 
fect circle,  and  its  axis  perpendicular  to  it.  Eut  the 


96  Of  the  Equation  of  Time.  Sec.  8 

earth's  diurnal  motion  on  an  inclined  axis,  and  its  an- 
nual motion  in  an  elliptical  orbit,  cause  the  Sun's  mo- 
tion in  the  Heavens  to  be  unequal  ;  for  sometimes  he 
revolves  from  the  meridian  to  the  meridian  again  in 
somewhat  less  than  24  hours ;  shewn  by  a  well  regu- 
lated clock,  and  at  other  times  in  somewhat  more  ;  so 
that  the  time  shown  by  a  true  going  clock,  and  true 
Sun-dial  is  never  the  same  ;  except  on  the  15th  day  of 
April,  the  16th  of  June,  the  31st  of  August,  and  the 
24th  day  of  December,  The  clock,  if  it  goes  equally 
true  during  the  whole  year,  will  be  before  the  Sun 
from  the  24th  of  December  till  the  15th  of  April;  from 
that  time  till  the  16th  of  June,  the  Sun  will  be  faster 
than  the  clock. 

The  point  where  the  Sun  is  at  his  greatest  distance 
from  the  earth  is  called  the  Sun's  apogee.  The  point 
where  he  is  at  his  least  distance  from  the  earth  is  cal- 
led his  perigee  ;  and  a  straight  line  drawn  through  the 
earth's  centre  from  one  of  those  points  to  the  other  is 
called  the  line  of  the  apsides. 

The  distance  that  the  Sun  has  gone  in  any  time  from 
his  apogee,  is  called  his  mean  anamoly,  and  is  reckoned 
in  signs,  degrees,  minutes  and  seconds,  allowing  30 
degrees-to  a  sign. 

OF  THE  PRECESSION  OF  THE  EQUINOXES. 

It  has  been  observed,  that  by  the  earth's  motion  on 
its  axis,  there  is  more  matter  accumulated  around  the 
equatorial  parts  than  any  where  else,  on  the  surface 


Sec.~8  Of  the  Precession  of  the  Equinoxes.          91 

of  the'earth.  The  Sun  and  Moon,  by  attracting  this 
redundancy  of  matter,  bring  the  equator  sooner  under 
them  in  every  return  towards  it,  than  if  there  were  no 
such  accumulation.  Therefore  if  the  Sun  sets  out  as 
from  any  star,  or  other  fixed  point  in  the  Heavens,  the 
moment  when  he  is  departing  from  the  equatorial,  or 
from  either  tropic,  lie  will  come  to  the  same  equinox  or 
tropic  again  20  minutes  and  17  and  J  seconds  of  time, 
or  50  seconds  of  a  degree,  before  he  completes  his 
course  so  as  to  arrive  at  the  same  fixed  star,  or  point 
from  whence  he  set  out.  For  the  equinoxial  points  re- 
cede 50  seconds  of  a  degree  westward  every  year,  con- 
trary to  the  Sun's  annual  progessive  motion. 

When  the  sun  arrives  at  the  same  equinoxial,*  or  sol- 
stitial point,  he  finishes  what  is  called  the  tropical 
year ;  which  by  observation  is  found  to  contain  365 
days,  5  hours,  48  minutes,  and  47  seconds,  and  when 
he  arrives  at  the  same  fixed  star  again,  as  seen  from  the 
earth,  he  completes  the  sydereal  year,  which  contains 
365  days,  6  hours,  9  minutes  14  and  J  seconds. 

The  sydereal  year  is  therefore  30  minutes  17  and  J 
seconds  longer  than  the  solar,  or  tropical  year,  and  9 
ininutes,14  and  J  seconds  longer  than  the  Julian  or  civil 
year;  which  we  state  at  365  days,  6  hours. 

As  the  Sun  describes  the  whole  ecliptic,  or  360  de- 


*  The  two  opposite  points  in  which  the  ecliptic  crosses  the  equinox 
are  called  the  equinoxial  points,  and  the  two  points  where  the  ecliptic 
touches  the  tropics,  (which  are  likewise  opposite,  and  90  degrees  from 
the  tropic,)  are  called  the  solstitial  points. 

L 


^^ 

98  Of  the  Precession  of  the  Equinoxes.          Sec.  8 

grees  in  a  tropical  year,  he  moves  59  minutes,  8  seconds 
of  a  degree  every  day,  at  a  mean  rate  ;  therefore  he  will 
arrive  at  the  same  equinox  or  solstice,  when  he  is  50  sec- 
onds of  a  degree  short  of  the  same  star  or  fixed  point 
in  the  Heavens,  from  which  he  set  out  in  the  year  be- 
fore. So  that  with  respect  to  the  fixed  stars,  the  Sun 
and  equinoxial  points  fall  back  30  degrees  in  2,160 
years,  which  wdll  make  the  stars  appear  to  have  gone 
30  degrees  forward  with  respect  to  the  signs  of  the  eclip- 
tic in  that  space  of  time  ;  for  the  same  signs  always 
keep  in  the  same  points  of  the  ecliptic,  without  regard 
to  the  constellations. 

The  Julian  year  exceeds  the  solar  by  11  minutes 
and  3  seconds,  which  in  1,438  years  amount 
to  eleven  days,  and  so  much  our  seasons  had  fallen 
back  with  respect  to  the  days  of  the  months  since  the 
time  of  the  Nicene  Council,  in  A.  D.  325,  and  therefore 
to  bring  back  all  the  feasts  and  festivals  to  the  days  then 
settledj  it  was  requisite  to  suppress  11  nominal  days.*-  - 
And  that  the  same  seasons  might  be  kept  to  the  same 
time  of  the  year,  for  the  future  to  leave  out  the  bissex- 
tile day  in  February,  at  the  end  of  every  century,  not 
dlvis  bleby  four,  reckoning  them  only  common  years,  as 
the  17th,  16th,  and  19th  centuries;  namely,  the  years 
1700,  1800,  and  1900,  &c.  because  a  day  intercalated 
every  fourth  year  was  too  much,  and  retaining  the  bis- 
sextile at  the  end  of  those  centuries  of  years  which  are 


*  The  difference  in  the  present  century,  between  the  old  and  new  styles,. 
in  twelve  days. 


Sec.  8  Of  the  Precession  of  the  Equinoxes.  •        99 

divisible  by  four,  as  the  years  1600,  2000,  2400,  &c. 
otherwise  in  length  of  time,  the  seasons  would  be  quite 
reversed  with  regard  to  the  months  of  the  year  ;  though 
it  would  have  required  near  23,783  years  to  have 
brought  about  such  a  total  change.  If  the  earth  had 
exactly  made  365  and  J  diurnal  revolutions  on  its  axis 
whilst  it  revolved  from  any  equinoxial  or  solstitial  point 
to  the  same  again,  the  civil  and  solar  years  would  al- 
ways have  kept  pace  together,  and  the  style  would 
neverhave  needed  any  alteration. 


100  Interrogations  for  Section  Eighth.          Sec.  8 


Interrogations  for  Section  Eighth. 

In  what  time  do  the  stars  appear  to  go  round  the 
Earth? 

In  what  time  does  the  Sun  appear  to  go  round  the 
Earth  ? 

In  what  time  do  the  stars  gain  one  revolution  1 

How  many  days  in  a  solar  year  ? 

How  many  in  a  sydereal  ? 

Is  the  motion  of  the  earth  on  its  own  axis  uniform  at 
all  times  of  the  year  1 

Are  the  sydereal  days  always  of  the  same  length  ? 

Is  the  Sun's  apparent  diameter  in  the  Heavens  always 
equal  ? 

On  what  days  of  the  year  are  the  Sun  and  clock 
together  ? 

Between  what  periods  will  the  clock  be  before  the 
Sun? 

Between  what  periods  will  the  Sun  be  before  the 
clock  ? 

What  is  called  the  Sun's  apogee? 

What  his  perigee  1 

What  the  line  of  the  Apsides  ? 

What  is  meant  by  the  Sun's  mean  anamoly  1 


Sec.  8  Interrogations  for  Section  Eighth.  101 

How  is  his  mean  anamoly  reckoned  ? 

What  is  meant  by  the  precession  of  the  equinoxes  1 

Is  there  more  matter  accumulated  at  the  equator  than 
at  any  other  part  oi  the  earth  ? 

What  is  the  cause  of  such  accumulation  ? 

What  is  the  Equator  1 

What  effect  is  produced  by  this  accumulation  of 
matter  ? 

How  many  seconds  of  a  degree  do  the  equinoxial 
points  recede  westward  every  year  ? 

What  are  meant  by  the  equinoxial  points  1 

What  the  solstitial  ? 

Which  is  the  longest,  the  sydereal  cr  solar  year,  and 
how  much  ? 

Howr  many  degrees  will  the  equinoxial  points  fall 
back  in  2,160  years  1 

Do  the  same  signs  always  keep  in  the  same  points  of 
the  ecliptic  ? 

Which  is  the  longest,  the  Julian  or  the  solar  year,  and 
how  much  ? 

How  many  days  difference  will  this  make  in  1,433 
years  ? 

In  wrhat  year  of  the  Christian  era  was  the  Council  of 
Nice  held? 

What  centuries  were  to  be  leap  years  ? 

What  is  the  difference  between  the  old  and  new  styles 
in  the  present  century? 


SECTION  NINTH. 
OF  TMJ3  JtfOOWS  PHASES. 

BY  looking  at  the  Moon  with  an  ordinary  telescope, 
we  perceive  that  her  surface  is  diversified  with  long 
tracts  of  prodigious  high  mountains,  and  dark  cavities. 
This  ruggedness  of  the  Moon's  surface  is  of  great  use 
to  us,  by  reflecting  the  Sun's  light  to  all  sides  ;  for  if 
the  Moon  were  smooth  and  polished,  she  could  never 
distribute  the  Sun's  light  all  around.  In  some  positions 
she  would  shew  us  his  image  no  larger  than  a  point, 
but  with  such  lusture  as  would  be  hurtful  to  our  eyes. 

The  Moon's  surface  being  so  uneven,  many  have 
wondered  why  her  edge  does  not  appear  jagged,  as 
well  as  the  curve,  bounding  the  light  and  the  dark  pla- 
ces. But  if  we  consider,  that  what  we  call  the  edge  of 
the  Moon's  disk  is  not  a  single  line,  set  round  with 


Sec.  9  Of  the  Moon's  Phases.  103 

mountains,  in  which  case  it  would  appear  irregularly 
indented,  but  a  large^zone,  having  many  mountains  ly- 
ing behind  each  other  from  the  observer's  eye,  we 
shall  find  that  the  mountains  in  some  rows  will  be  op- 
posite to  the  valves  in  others,  and  and  so  fill  up  the  in- 
equalities, as  to  make  her  appear  quite  round. 

The  Moon  being  an  opaque  spherical  body,  (for  her 
hills  take  off  no  more  of  her  roundness,  than  the  ine- 
qualities on  the  surface  of  an  orange  take  off  from  its 
roundness,)  we  can  only  see  that  part  of  her  enlight- 
ened half  which  is  towards  the  earth.  Therefore, 
when  she  is  in  conjunction  with  the  Sun,  her  dark  half 
is  towards  the  earth,  and  she  disappears  ;  there  being 
no  light  on  that  part  to  render  it  visible.  When  she 
comes  to  her  first  octant,  or  has  gone  over  one  eighth 
part  of  her  orbit  from  her  conjunction,  a  quarter  of  her 
enlightened  side  is  seen  towards  the  earth,  and  she 
appears  horned.  When  she  has  gone  a  quarter  of  her 
orbit  from  between  the  earth  and  Sun  ;  she  shows  us 
one  half  of  her  enlightened  side,  and  then  she  is  said  to 
be  a  quarter  old.  When  she  has  gone  another  octant, 
she  shows  us  more  of  her  enlightened  side,  and  then 
she  appears  gibbous  ;  and  when  she  has  gone  over  half 
her  orbit,  her  whole  enlightened  side  is  towards  the 
earth,  and  therefore  she  appears  round  :  we  then  say 
it  is  full  Moon,  or  the  Moon  is  in  opposition  with  the 
Sun.  In  her  third  octant,  part  of  her  dark  side  being 
towards  the  earth,  she  again  appears  gibbous,  and  is  on 
the  decrease.  In  her  third  quarter,  she  appears  half 
decreased.  When  in  her  fourth  octant,  she  again  ap- 


104  Of  the  Mom's  Phases.  Sec.  9 

pears  horned.  And  after  having  completed  her  course 
from  the  Sun,  to  the  Sun  again,  she  disappears,  and  we 
say  it  is  new  Moon.  But  when  she  is  seen  from  the 
Sun,  she  appears  always  full. 

The  Moon's  absolute  motion  from  her  change  to  her 
first  quarter  is  so  much  slower  than  the  earth's  that 
she  falls  240  thousand  miles  (equal  to  the  semi-diame- 
ter of  her  orbit)  behind  the  earth  at  her  first  quarter, 
that  is,  she  falls  back  a  space  equal  to  her  distance 
from  the  earth.  From  that  time  her  motion  is  gradu- 
ally accelerated  to  her  opposition  or  full,  and  then  she 
is  come  up  as  far  as  the  earth,  having  regained  what 
she  lost  in  her  first  quarter,  her  motion  continues  ac- 
celerated so  as  to  be  just  as  far  before  the  earth  as  she 
was  behind  it  at  her  first  quarter.  But  from  her  third 
quarter  her  motion  is  so  retarded,  that  she  loses  as 
much  with  respect  to  the  earth,  as  is  equal  to  her  dis- 
tance from  it,  or  to  the  semi-diameter  of  her  orbit,  and 
by  that  means  the  earth  comes  up  with  her,  and  she 
is  again  in  conjunction  with  the  sun  as  seen  from  the 
earth.  Hence  we  we  find  that  the  moon's  absolute 
motion  is  slower  than  the  earths  from  her  third  quar- 
ter to  her  first,  and  swifter  than  the  earth's  from  her 
first  quarter  to  her  third  ;  her  path  being  less  curved 
than  the  earth's  in  the  former  case,  and  more  in  the 
latter.  Yet  it  is  still  bent  the  same  way  towards  the 
sun ;  for  if  we  imagine  the  concavity  of  the  earth's 
orbit  to  be  measured  by  the  length  of  a  perpendicular 
line  let  down  from  the  earth's  place  upon  a  straight 
line  at  the  full  of  the  moon,  and  connecting  the  pla- 


Sec.  9  Of  the  Moon's  Pluirsss.  105 

ces  of  the  earth  at  the  end  of  the  moon's  first  and 
third  quarters  ;  that  length  will  be  about  640,000  miles, 
and  the  moon  when  new  only  approaching  nearer  to 
the  sun  by  240,000  miles  than  the  earth.  The  length 
of  the  perpendicular  line  let  down  from  her  place  at 
that  time  upon  the  same  straight  line,  and  which  shows 
the  concavity  of  that  part  of  her  path  will  be  about 
400,000  miles. 

The  moon's  path  being  concave  to  the  sun  through- 
out, demonstrates  that  her  gravity  towards  the  sun  at 
the  time  of  her  conjunction,  exceeds  her  gravity  to- 
wards the  earth.  And  if  we  consider  that  the  quan 
tity  of  matter  in  the  sun,  is  nearly  230  thousand  times 
as  great  as  the  quantity  of  matter  in  the  earth,  and 
that  the  attraction  of  each  body  diminishes  as  the 
squares  of  their  distances  from  each  other  increase, 
we  shall  soon  find  that  the  point  of  equal  attraction 
between  the  earth  and  the  sun  is  about  70  thousand 
miles  nearer  the  earth,  than  the  moon  is  at  her  change. 
It  may  then  appear  surprising  that  the  moon  does  not 
abandon  the  earth  when  she  is  between  it  and  the  sun, 
for  she  is  considerably  more  attracted  by  the  sun,  than 
by  the  earth  at  that  time.  But  this  difficulty  vanishes 
when  we  discover  that  a  common  impulse  on  any  sys- 
tem of  bodies  affects  not  their  relative  motions  ;  but 
that  they  will  continue  to  attract,  impel  or  circulate 
round  one  another  in  the  same  manner,  as  if  there 
were  no  such  impulse.  The  moon  is  so  near  the  earth, 
and  both  of  them  so  far  from  the  sun,  that  the  attrac- 
tive power  cf  the  sun  may  be  considered  as  equal  on 


M 


106  Of  the  Moon's  Phases.  Sec.  9 

both.  Therefore  the  moon  will  continue  to  circulate 
round  the  earth  in  the  same  manner,  as  if  the  sun  did 
not  attract  them  at  all. 

OF  THE  PHENOMENA  OF  THE  HARVEST  MOON. 

It  is  generally  believed  that  the  Moon  rises  about  50 
minutes  later  every  day,  than  on  the  preceding  ;  but 
this  is  true  only  to  places  on  the  equator.  In  places 
of  considerable  latitude,  there  is  a  remarkable  differ- 
ence, especially  in  the  time  of  the  autumnal  harvest, 
with  which  farmers  were  formerly  better  acquainted 
than  Astronomers. 

In  this  instance  of  the  Harvest  Moon,  as  in  many 
others  discoverable  by  Astronomy,  the  wisdom  and 
beneficence  of  the  Deity  is  conspicuous,  who  really  or- 
dered the  course  of  the  Moon,  so  as  to  bestow  more  or 
less  light  on  all  parts  of  the  earth,  as  their  several  cir- 
cumstances and  seasons  render  it  more  or  less  ser- 
viceable. 

About  the  equator,  where  there  is  no  variety  of  sea- 
sons, and  the  weather  seldom  changes,  except  at  sta- 
ted times  ;  moonlight  is  not  necessary  for  gathering 
the  produce  of  the  ground,  and  there  the  Moon  rises 
about  50  minutes  later  every  day  or  night,  than  on  the 
former. 

In  considerable  distances  from  the  equator,  where 
the  weather  and  seasons  are  more  uncertain,  the  au- 
tumnal full  Moons  rise  very  soon  after  Sun-set,  for 
several  evenings  together.  At  the  polar  circles,  where 


Sec.  9  Of  the  Moon's  Phases.  107 

the  mild  season  is  of  very  short  duration,  the  autumnal 
full  Moon  rises  at  Sun-set  from  the  first  to  the  third 
quarter:  and  at  the  poles,  where  the  Sun  is  during 
half  the  year  absent,  the  winter  full  Moons  shine  con- 
stantly without  setting  from  the  first  to  the  third 
quarter. 

It  is  evident  that  all  these  phenomena  are  owing  to 
the  different  angles  made  by  the  horizon,  and  different 
parts  of  the  Moon's  orbit,  and  that  the  Moon  can  be  full 
but  once  or  twice  in  a  year,  in  those  parts  of  her  orbit 
which  rise  with  the  least  angles. 

The  plane  of  the  Equinox  is  perpendicular  to  the 
earth's  axis,  and  therefore  as  the  earth  turns  round  in 
its  diurnal  revolution,  all  parts  of  the  Equinox  make 
equal  parts  with  the  horizon,  both  at  rising  and  set- 
ting ;  so  that  equal  portions  of  it  always  rise  or  set  in 
equal  times.  Consequently  if  the  Moon's  motions 
were  equible,  and  in  the  equinox  at  the  rate  of  12  de- 
grees and  1 1  minutes  from  the  Sun  every  day,  as  it  is 
in  her  orbit ;  she  would  rise  and  set  50  minutes  later 
every  day  than  on  the  preceding  ;  for  12  degrees,  11 
minutes  of  the  equator  rise  or  set  in  50  minutes  of 
time  in  all  latitudes. 

The  different  parts  ef  the  ecliptic,  on  account  of  its 
obliquity  to  the  earth's  axis,  make  very  different  an- 
gles with  the  horizon,  as  they  rise  or  set.  These  parts 
or  signs,  which  rise  with  the  smallest  angles,  set  with 
the  greatest,  and  rise  vice  versa,  In  equal  times, 
whenever  this  angle  is  least,  a  greater  portion  of  the 
ecliptic  risesj  than,  when  the  angle  is  larger,  as  may  be 


108 


Of  tht  Moon's  Phases. 


Sec.  9 


seen  by  elevating  the  pole 

I    CL 

51 

of  a  common  globe   to  any 

9 

-                         & 

If: 

IS 

considerable    latitude,  and 

1                         ^ 

11' 

re  ST. 

3    3 

o  aq 

then  turning  it  round  on  its 

n 

M 

CD 

? 

own  axis  in  the  horizon.  — 

hm 

h     m 

Consequently,      when    the 

1  Cancer,    1311     5 

0  50 

Moon  is  in  those  signs,which 

2 

26 

1    10 

0  43 

rise  or  set  with  the  smallest 

3 

Leo,          10 

1    14 

0  37 

angles,  she  rises  or  sets  with 

4 

23 

1    17 

0  32 

the  least  difference  of  time, 

5 

Virgo,        6 

1    16 

0  28 

and  with  the  greatest  differ- 

6 

19 

1    15 

0  24 

ence  in    those  signs  which 

7 

Libra,         2 

1    15 

0  20 

rise  or  set  with  the  greatest 

8 

15 

1    15 

0   18 

angles.    On  the  parallel  of 

9 

28 

1    15 

0  17 

London,  as    much   of.  the 

10 

Scorpio,   12 

1    15 

0  22 

ecliptic  rises  about  Pisces 

11 

25 

1    14 

0  30 

and  Aries  in  two  hours,  as!  ^ 

Sagitarius,8 

1    13 

0  39 

the  Moon  goes  through  in 

13 

21 

1    10 

0  47 

six  days,  &/  therefore,  when 

14 

Capricorn,  4 

1      4 

0  56 

the  Moon  is  in  these  signs, 

15 

17 

0  46 

1     5 

she  differs  but  two  hours  inj1^ 

Aquarius,  1 

0  40 

1     8 

rising  for  six  days  together, 

17 

14 

0  35 

1   12 

that  is  about  20  minutes  la- 

18 

27 

0  30 

1    15 

ter  every  day  or  night,  than 

19 

Pisces,      10 

0  25 

1    16 

on  the  preceding  at  a  mean 

20 

23 

0  20 

1    17 

rate.     But  in  14  days  after- 

21 

Aries,         7 

0  17 

1    16 

wards  the  Moon  comes  to 

22 

20 

0   17 

1   15 

Virgo  and  Libra,  which  are 

23 

Taurus,      3 

0  20 

1    15 

the  opposite  signs  to  Pisces 

24 

16 

0  24 

I    15 

and  Aries,  and  then  she  dif- 

25 

29 

0  30 

1    14 

fers  almost  four   times   as 

26 

Gemini,    13 

0  40 

1    13 

much  in   rising  namely  : 
one  hour  and  about  15  min- 

27 
28 

26 
Cancer,      9 

0  56 
1   00 

1     7 

1  58 

utes  later  every  day  or  night  than  the  former,  whilst 
she  is  in  these  signs.  The  annexed  table  shows  the 
daily  mean  difference  of  the  Moon's  rising  and  setting 


Sec.  9  Of  the  Moon's  Phases.  109 

on  the  parallel  of  London,  for  28  days,  in  which  time 
the  Moon  finishes  her  period  round  the  ecliptic,  &  gets 
9  degrees  into  the  same  sign  from  the  beginning  of 
which  she  set  out.  It  appears  by  the  table,  that  when 
the  Moon  is  in  Virgo  and  Libra,  she  rises  one  hour  and 
a  quarter  later  every  day  than  she  rose  on  the  former, 
and  differs  only  28,  24,. 20,  18,  or  17  minutes  in  setting. 
Bat  when  she  comes  to  Pisces  and  Aries,  she  is  only 
20  or  17  minutes  later  in  rising. 

In  the  time  that  the  Moon  goes  round  the  ecliptic 
from  any  conjunction  or  opposition,  the  earth  goes  al- 
most a  sign  forward,  and  therefore  the  Sun  will  seem 
to  go  as  far  forward  in  that  time  ;  (namely  27  and  -J 
degrees,)  so  that  the  Moon  must  go  27  and  i  degrees 
more  than  round,  and  as  much  farther  as  the  Sun  ad- 
vances in  that  interval;  which  is  2  degrees  and  }5  be- 
fore she  can  be  in  conjunction  with,  or  opposite  to  the 
Sun  again.  Hence,  it  is  evident,  that  there  can  be  but 
one  conjunction,  or  opposition  to  the  Sun  and  Moon,  in 
any  particular  part  of  the  ecliptic  in  the  course  of  a 
year. 

As  the  Moon  can  never  be  full  but  when  she  is  oppo- 
site to  the  Sun,  and  the  Sun  is  never  in  Virgo  and  Li- 
brn  only  in  our  autumnal  months,  it  is  plain  that  the 
Moon  is  never  full  in  the  opposite  signs,  Pisces  and 
Aries,  but  in  those  two  months.  Therefore  we  can 
have  only  two  full  Moons  in  the  year,  which  rise  so 
near  the  time  of  setting  for  a  week  together,  as  above 
mentioned.  The  former  of  these  is  called  the  Harvest 
Moon,  the  latter  the  Hunter's  Moon. 


110  Of  the  Moon's  Phases.  Sec.  9 

In  northern  latitudes,  the  autumnal  full  Moons  are  in 
Pisces  and  Aries ;  and  the  vernal  full  Moons  in  Virgo 
and  Libra,in  southern  latitudes  just  the  reverse  because 
the  seasons  are  contrary.  But  \irgo  and  Libra  rise, 
at  as  small  angles  with  the  horizon  in  southern  latitudes, 
as  Pisces  and  Aries  do  in  the  northern  j  and  therefore 
the  Harvest  Moons  are  just  as  regular  on  one  side  of 
the  equator,  as  on  the  other. 

As  these  signs  which  rise  with  the  least  angles,  set 
with  the  greatest ;  the  several  full  Moons  differ  as  much 
in  their  times  of  rising  every  night,  as  the  autumnal  full 
Moons  differ  in  their  times  of  setting  ;  and  set  with  as 
little  difference  as  the  autumnal  full  Moons  rise  ;  the 
one,  being  in  all  cases  the  reverse  of  the  other. 

For  the  sake  of  plainness,  the  Moon  has  been  suppo- 
sed to  move  in  the  ecliptic  from  which  the  Sun  never 
deviates.  But  the  orbit  in  which  the  Moon  really  moves 
is  different  from  the  ecliptic,  one  half  being  elevated  5 
and  i  degrees  above  it,  and  the  other  half,  as  much  de- 
pressed below.  The  Moon's  orbit  therefore,  intersects 
the  ecliptic  in  two  points  diametrically  opposite  to  each 
other,  and  these  intersections  are  called  the  Moon's 
Nodes.  The  Moon  can  therefore  never  be  in  the  eclip- 
tic but  when  she  is  in  either  of  her  Nodes,  which  is  at 
least  twice  in  every  course,  and  sometimes  thrice.  For 
as  the  Moon  goes  almost  a  whole  sign  more  than  round 
her  orbit,  from  change  to  change,  if  she  passes  by  either 
node  about  the  time  of  her  conjunction,  she  will  pass 
by  the  other  in  about  fourteen  days  after,  and 
came  round  to  the  former  node  two  days  again  be- 


Sec.  9  Of  the  Moon's  Phases.  1 1 1 

fore  the  next  change.  That  node  from  which  the 
Moon  begins  to  ascend  northwardly,  or  above  the  eclip- 
tic in  northern  latitudes,  is  called  the  ascending  node, 
and  the  other  the  descending  node  ;  because  the  Moon 
when  she  passes  by  it,  descends  below  the  ecliptic  south- 
ward. 

The  Moon's  oblique  motion  with  regard  to  the  eclip- 
tic, causes  some  difference  in  the  times  of  her  rising 
and  setting  from  that  which  has  been  already  men- 
tioned. 

When  she  is  northward  of  the  ecliptic,  she  rises 
sooner,  and  sets  later,  than  if  she  moved  in  the  ecliptic; 
and  when  she  is  southward  of  the  ecliptic,  she  rises 
later  and  sets  sooner.  This  difference  is  variable  even 
in  the  same  signs,  because  the  nodes  shift  backward 
about  19  degrees  and  §  in  the  ecliptic  every  year ;  and 
so  go  round  it  contrary  to  the  order  of  signs  in  18  years 
and  225  days. 

When  the  ascending  node  is  in  Aries,  the  southern 
half  of  the  Moon's  orbt,  makes  an  angle  of  5  and  J  de 
grees  less  with  the  horizon,  than  the  ecliptic  does, 
when  Aries  rises  in  northern  latitudes  :  for  this  rea- 
son, the  Moon  rises  with  less  difference  of  time,  while 
she  is  in  Pisces  &,  Aries,  than  she  wrould  do  if  she  kept 
in  the  ecliptic.  But  in  9  years  and  112  days  after- 
ward,the  descending  node  comes  to  Aries,  and  then  the 
Moon's  orbit  makes  an  angle  of  5  and  |  degrees  great- 
er with  the  horizon  when  Aries  rises,  than  the  ecliptic 
docs  at  that  time  ;  which  causes  the  Moon  to  rise  with 


112  Of  the  Moon's  Phases.  Sec.  9 

greater  difference  of  times  in   Pisces  and  Aries,  than 


if  she  moved  in  the  ecliptic. 

When  the  ascending  node  is  in  Aries,  the  angle  is 
only  9  and  j  degrees  on  the  parallel  of  London  when 
Aries  rises.  But  when  the  descending  node  comes  to 
Aries,  the  angle  is  20  and  J  degrees  ;  this  occasions  as 
great  difference  of  the  Moon's  rising  in  the  same  signs 
every  9  years,  as  these  would  be  on  two  parallels  10 
and  §  degrees  from  each  other,  if  the  Moon's  course 
were  in  the  ecliptic.  The  following  table  shows  how 
much  the  obliquity  of  the  Moon's  orbit,  affects  her  ri- 
sing and  setting  on  the  parallel  of  London,  from  the 
12th  to  the  18th  day  of  her  age,  supposing  her  to  be 
full  at  the  autumnal  equinoxes,  and  then  either  in  the 
ascending  node,  or  (highest  part  of  her  orbit,)  and  in 
the  descending  node,  (or  least  part  of  her  orbit.)  M 
signifies  morning.  A  afternoon,  and  the  line  at  the 
foot  of  the  table  shows  a  week's  difference  in  rising  and 
setting. 


Full  in  her 
ascending  node. 

Full  in  the   highest 
part  of  her  orbit 

Full  in  her 
descending   node 

Full   in   the    lowest 
part  of  her  orbit 

7noons 
age 

Rises    at 

H                 M 

Sets  at 

H                M 

Rises    at- 

H            ?,: 

Sets    at 

H                M 

Rises    at 

H                M 

Sets    at 

H                  JVI 

Rises     at 

H                 X 

Sets  at 

H                 M 

12 
13 

14 
15 
16 
17 

18 

5  A  15 
5        32 

5        48 
6          5 
6        20 
6        36 
6        54 

3  M  20 
4        25 
5        30 
7         0 
8       15 
9        12 
10      30 

4  A  30 
4        50 
5        15 
5        42 
6          2 
6        26 
7          0 

3  M  15 
4        45 
6           0 
7        20 
8        35 
9         45 
10       40 

4  A  32 
5        15 
5        45 
6        15 

6       46 
7        18 
8         0 

3  M  40 
4        20 
5        40 
6        56 
8          0 
9        15 
10      20 

5  A  16 
6          0 
6        20 
6        45 
7          8 
7        30 
7        52 

3  M     0 
4         15 
5        28 
6        32 
7        45 
9        15 
10        0 

differ 
ejice. 

1        39 

7        10J2        30J7        25 

3        28  6        40 

2        36 

7          0 

This  table  was  not  computed,  but  only  estimated,  as 
near  as  could  be  done  from  a  common  globe,  on  which 


Sec.  9  Of  the  Moon's  Phases.  113 

the  Moon's  orbit  was  delineated  with  a  black   lead 
pencil. 

As  there  is  a  complete  revolution  of  the  nodes  in 
18  J  years,  there  must  be  a  regular  period  of  all  the 
varieties  which  can  happen  in  the  rising  and  setting  of 
the  Moon  during  that  time. 

At  the  polar  circles,  when  the  Sun  touches  the  Sum- 
mer tropic,  he  continues  twenty-four  hours  above  the 
horizon  ;  and  the  like  number  below  it,  when  he  touch- 
es the  Winter  tropic.  For  the  same  reason,  the  full 
Moon  being  as  high  in  the  ecliptic  as  the  Summer's 
Sun,  must  therefore  continue  as  long  above  the  hori- 
zon :  and  the  Summer  full  Moon  being  as  low  in  the 
ecliptic  as  the  Winter  Sun,  can  no  more  rise  than  he 
does.  But  these  are  only  the  two  full  Moons  which 
happen  about  the  tropics,  for  all  the  others  rise  and 
set.  In  Summer,  the  full  Moons  are  low,and  their  stay 
is  short  above  the  horizon,  then  the  nights  are  short, 
and  we  have  the  least  occasion  for  Moon-light.  In 
Winter,  the  full  Moons  run  high,  and  they  stay  long 
above  the  horizon  when  the  nights  are  long,  and  we 
need  the  greatest  quantity  of  her  reflected  light. 

At  the  poles,  one  half  of  the  ecliptic  never  sets,  and 
the  other  half  never  rises  ;  and  therefore  as  the  Sun  is 
always  half  a  year  in  describing  one  half  of  the  eclip- 
tic, and  as  long  in  going  through  the  other  half,  it  is 
natural  to  imagine  that  the  Sun  continues  half  a  year 
together  above  the  horizon  of  each  pole  in  its  turn, 
and  as  long  below  it ;  rising  to  one  pole  when  he  sets 
to  the  other,  This  would  be  exactly  the  case,  if  there 


114  Of  the  Moon's  Phases.  Sec.  9 

were  no  refraction.  But  by  the  refraction  of  the  Sun's 
rays,  occasioned  by  the  atmosphere,  he  becomes  visi- 
ble some  days  sooner,  and  continues  some  days  longer 
in  sight,  than  he  would  otherwise  do:  so,  that  he  ap- 
pears above  the  horizon  of  either  pole,  before  he  has 
got  below  the  horizon  tf  the  other. 

And  as  he  never  goes  more  than  23  degrees  and  28 
minutes  below7  the  horizon  of  the  poles,  they  have  very 
little  dark  night ;  twilight  bein^  there,  as  well  as  at 
all  other  places,  till  the  Sun  be  13  degrees  below  the 
horizon.  The  full  Moon  being  always  opposite  to  the 
Sun,  can  never  be  seen  while  the  Sun  is  above  the  ho- 
rizon, except  when  the  Mo  n  falls  in  the  northern  half 
of  her  orbit ;  for  when  any  point  of  the  ecliptic  rises, 
the  opposite  point  sets.  Therefore,  as  the  Sun  is 
above  the  horizon  of  the  north  pole  from  the  20th  of 
March,  till  the  23  J  September;  it  is  plain  that  the 
Moon  when  full,  being  ( j>] osi'e  to  the  Sun,  must  be 
below  the  horizon  during  that  half  of  the  year.  But 
when  the  Sun  is  in  the  southern  half  of  the  ecliptic,  he 
never  rises  to  the  north  pole  ;  during  this  half  of  the 
year,  every  full  Moon  happens  in  seme  part  of  the 
northern  half  of  the  ecliptic  which  never  sets.  Conse- 
quently, as  the  polar  inhabitants  never  see  the  full 
Moon  in  Summer,  they  have  her  always  in  the  Win- 
ter ;  before,  at,  and  after  the  fuP,  shining  during  14  of 
our  days  and  nights.  And  when  the  Sun  is  at  his 
greatest  distance  bekw  the  horizon,  being  then  in 
Capricorn,  the  Mcon  is  at  !  er  first  quarter  in  Aries, 
full  in  Cancer,  and  at  her  third  quarter  in  Libra.  And 


Sec.  9  Of  the  Moon's  Phases.     *,  J 1 5 

as  the  beginning  of  Aries  is  the  rising  point  of  the  eclip- 
tic, Canrcr  the  highest,  ami  Libra  UK*  setting  point,  the 
M-.ioii  ris  'S  at  her  first  quarter  in  Aries,  is  most  elevated 
.*!» >w  me  horiz  >n,  an;i  hi  I  in  Cancer,  and  sets  at  the 
beginning  of  Libra  in  iu  r  third  qsiartcr,  having  eontin- 
ui'tl  visible  for  14  iliurnnl  rotations  of  the  earth.  Thus 
the  poles  are  supplied  one  half  of  the  Wintertime  with 
constant  M  >o!i-light,  in  the  absence  of  the  bun,  and 
only  lose  sight  of  the  Moon  from  her  third  to  her  first 
quarter,  while  she  gives  but  very  liulc  light,  and  could 
be  but  of  little,  and  sometimes  of  no  service  to  them. 


110  Interrogations  for  Section  J\finth.          Sec.  9 


Interrogations  for  Section  Xinth. 

What  is  understood  by  the  MOON'S  PHASES  1 

What  is  discovered  by  observing  the  moon  with  a 
telescope  ? 

Of  what  use  is  the  ruggedness  to  us  ? 

If  the  surface  of  the  moon  is  uneven,  why  does  it  not 
so  appear  when  viewed  by  the  eye  only  1 

What  is  the  moon  ? 

What  part  of  the  mo;m  do  we  discover  1 

When  is  she  said  to  be  in  conjunction  with  the  Sun  ? 

When  she  is  in  her  first  octant,  how  much  of  her  en- 
lightened side  is  visible  1 

How  much  of  her  enlightened  side  does  she  show  in 
her  first  quarter  ? 

When  she  has  gone  half  around  her  orbit,  how  does 
she  appear  ? 

How  does  she  appear  when  viewed  from  the  Sun  ? 

Are  the   moon's   morions  faster  or  slower  than  the 
earth's,  from  her  change  to  her  first  quarter  ? 

How  far  does  she  fall  behind  the  earth  ? 

'er  first  qu.utcT  to  her  full,  which  moves  with 
tv»i  rapidity  ? 


Sec.  9  Interrogations  for  Section  Ninth.  117 

Which  from  the  full  to  her  third  quarter  ? 

Which  from  the  third  quarter  to  the  change  1 

Is  the  gravity  of  the  moon  at  any  time  greater  towards 
the  Sun,  than  towards  the  earth,  and  at  what  time  ? 

How  much  greater  is  the_quantity  of  matter  in  the 
Sun,  than  in  the  earth  ? 

In  what  proportion  does  the  attraction  of  each  body 
diminish  ? 

How  far  from  the  earth  is  the  point  of  equal  attrac- 
tion between  the  earth  and  the  Sun  ? 

Why  does  not  the  moon  leave  the  earth  and  go  to  the 
Sun  ? 

What  is  understood  by  the  Harvest  Moon  ? 

How  many  minutes  later  at  the  equator  does  the 
moon  rise  every  day,  than  on  the  preceding  ? 

Is  there  any  material  difference  in  high  northern,  of 
southern  latitudes  1 

At  what  time  in  northern  latitudes,  does  the  full  moon 
rise? 

How  many  days  together  does  the  moon  in  such  cases 
rise  at  nearly  the  same  time  ? 

What  is  the  cause  of  this  small  difference  ? 

How  far  does  the  earth  advance  in  her  orbit,  while  the 
moon  goes  round  the  ecliptic  ? 

How  mnny  conjunctions  and  oppositions  of  the  Sun 
and  Moon  can  take  place  in  any  particular  part  of  the 
ecliptic,  in  the  course  of  a  year? 

How  many  full  Moons  in  the  course  of  a  year,  that 
rise  with  so  little  difference  near  the  time  of  Sun- 
setting  ? 


1 18  Interrogations  for  Section  Ninth.  Sec.  9 

Do  s  this  singularity  appear  \i-.  ."Oniht  rs:  >ntittlr,s  <\* 
\vcll  as  in  nott1  ern  ? 

Docs  the  M  on's  orbit  lie  ex.  e  ly  in  1 1  e  ec  ipiic  r 

Does  the  moon's  orhii  intersect  ihr-  ecliptic  ? 

What  is  understood  by  the  moon's  no:i<  s  ? 

How  many  times  from  change  to  change,  is  the  moon 
in  her  nodes  ? 

Which  is  called  the  ascending  node  ? 

Wiiich  is  called  the  descending  node? 

Ho\v  many  degrees  are  they  asunder? 


How  much  does  these  nodes  shift  in  ihc  course  of  a 
year  ? 

Which  way  do  they  shift  ? 

In  what  length  of  time  do  they  go  around  the  eclip- 
tic ? 

How  many  degrees  can  the  Sun  go  below  the  hori- 
zon of  the  poles  ? 

How  many  degrees  must  the  Sun  be  below  the  hori- 
zon before  the  twilight  is  wholly  gone  ? 

Is  the  full  moon  in  the  Summer  season  ever  seen  at 
the  north  pole  ? 

Is  it  cominually  seen  in  Winter,  from  the  first  to  her 
third  quarter  ? 

Is  it  the  same  at  the  south  pole  ? 


SECTION  TENTH. 
On  the  Ebbing  and  Flowing  of  the  Sea. 

THE  cause  of  the  tides  was  first  discovered  by  Kep- 
ler, whj  thus  explains  it.  The  orb  of  the  attracting 
power,  (which  is  in  the  Moon,)  is  extended  as  far  as 
the  earth,  and  draws  the  waters  under  the  torrid  zone; 
acting  upon  places  where  it  is  vertical  ;  insensibly  on 
confined  Seas  and  Bays ;  but  sensibly  on  the  Ocean, 
whose  beds  are  larger,  and  the  waters  have  the  liberty 
of  reciprocation,that  is  of  rising  and  falling.  And  in  the 
70th  page  of  his  lunar  Astroncmy  he  says :  But  the 
cause  of  the  Tides  of  the  Sea,  appears  to  be  the  bo- 
dies of  the  Si:n  and  Moon,  drawing  the  waters  of  the 
Sea. 

This  hint  being  g^ven,  the  immortal  Sir  Isaac  New- 
ton improved  it,  and  wrote  so  amply  on  the  subject  as 


120      On  the  Ebbing  and  Flowing  of  the  Sea.      Sec.  10 

to  make  the  theory  of  Tides  in  a  manner  quite  his  own, 
by  discovering  the  cause  of  their  rising  on  the  side  of 
the  earth  opposite  to  the  Moon.  For  Kepler  believed 
that  the  presence  of  the  Moon  occasioned  an  impulse 
which  caused  another  in  her  absence. 

It  has  been  already  mentioned,  that  the  power  of 
gravity  diminishes  as  the  square  of  the  distance  in- 
creases, and  therefore  the  waters  on  the  side  of  the 
earth,  next  the  Moon,  are  more  attracted  than  the  cen- 
tral parts  of  the  earth  by  the  Moon,  and  the  central 
parts  are  more  attracted  by  her,  than  the  waters  on  the 
opposite  side  of  the  earth.  Therefore  the  distance  be- 
tween the  earth's  centre,  and  the  water,  or  its  surface, 
__>vill  be  increased.  If  the  attraction  be  unequal,  then 
that  body  which  is  most  strongly  attracted,  will  move 
with  greater  rapidity,  and  this  will  increase  its  distance 
from  the  other  body. 

As  this  explanation  of  the  ebbing  and  flowing  of  the 
Sea  is  deduced  from  the  earths'  constantly  falling  to- 
ward the  Moon  by  the  po\ver  of  gravity,  some  may  find 
a  difficulty  in  conceiving  how  this  is  possible  when  the 
Moon  is  full,  or  in  opposition  to  the  Sun  ;  since  the 
earth  revolves  about  the  Sun,  and  must  continually  fall 
towards  it ;  and  therefore  cannot  fall  contrary  ways 
at  the  same  time  :  or  if  the  earth  is  constantly  falling 
towards  the  Moon,  they  must  come  together  at 
last.  To  remove  this  difficulty,  let  it  be  considered 
that  it  is  not  the  centre  of  the  earth  that  describes 
the  annual  orbit  round  the  Sun,  but  the  common 


Sec.  10       On  the  Ebbing  and  Flowing  of  the  Sea.      1 2 1 

centre  *  of  gravity  of  the  earth  and  Moon  together  ; 
and  that  while  the  earth  is  moving  round  the  Sun,  it 
also  describes  a  circle  around  that  centre  of  gravity, 
going  as  many  times  around  it  in  one  revolution  about 
the  Sun,  as  there  are  lunations,  or  courses  of  the 
Moon  around  the  earth;  is  constantly  falling  towards 
the  Moon  from  a  tangent  to  the  circle  it  describes* 
around  the  said  centre  of  gravity. 

The  influence  of  the  Sun  in  raising  the  Tides,  is  but 
small  in  comparison  of  the  Moon's  :  though  the  earth's 
diameter  bears  a  considerable  proportion  to  its  distance 
from  the  Moon,  it  is  next  to  nothing  when  compared  to 
its  distance  from  the  Sun.  Therefore,  the  difference 
of  the  Sun's  attraction  on  the  sides  of  the  earth,  under, 
and  opposite  to  him,  is  much  less  than  the  difference 
of  the  Moon's  attraction  on  the  sides  of  the  earth  un- 
der, and  opposite  to  her :  therefore  the  Moon  must 
raise  the  Tides  much  higher  than  they  can  be  raised  by 
the  Sun.  On  this  theory,  (so  far  as  it  has  been  ex- 
plained,) the  Tides  ought  to  be  the  highest  directly 
under,  and  opposite  to  the  Moon ;  that  is,  when  the 
Moon  is  due  north  or  south.  But  we  find  in  open 
Seas,  where  the  water  flows  freely,  the  Moon  is  gen- 


*  This  centre  is  as  much  nearer  the  earth's  centre,  than  the  Moons,  as 
the  earth  is  heavier,  or  contains  a  greater  quantity  of  matter  than  the  Moon 
which  is  about  40  times.  If  both  bodies  were  suspended  from  it,  they  would 
hang  in  equilibrio.  Therefore  divide  the  Moon's  distance  from  the 
earth's  centre, (240,000  miles,)  by  40,  and  the  quotient, will  be  the  distance 
from  the  centre  of  the  earth  to  the  centre  of  gravity,  which  is  6,000  miles,  or 
2,000  from  the  earth's  surface. 

O 


On  Tides.  Sec.  10 

erally  past  the  north  and  south  meridian,  when  it  is 
high  water.  The  reason  is  obvious,  were  the  Moon's 
attraction  to  cease  wholly  when  she  was  past  the  me- 
ridian, yet  the  motion  of  ascent  communicated  to  the 
water  before  that  time,  would  make  it  continue  to  rise 
for  some  time  afterward,  much  more  must  it  continue 
to  rise,  when  the  attraction  is  only  diminished.  A  lit- 
tle impulse  given  to  a  moving  ball,  will  cause  it  to 
move  further  than  it  otherwise  would  have  done.  Or, 
as  experience  shows  that  the  weather  in  Summer,  is 
warmer  at  2  o'clock  in  the  afternoon,  than  when  the 
Sun  is  on  the  meridian,  because  of  the  increase  made 
to  the  heat  already  imparted. 

The  Tides  do  not  always  answer  to  the  same  dis- 
tance of  the  Moon  from  the  meridian  at  the  same  pla- 
ces, but  are  variously  affected  by  the  action  of  the  Sun} 
which  brings  them  on  sooner  when  the  Moon  is  in  her 
first  and  third  quarters,  and  keeps  them  back  later 
when  she  is  in  her  second  and  fourth ;  because,  in  the 
one  case,  the  Tide  raised  by  the  Sun  alone,  would  be 
earlier  than  the  Tide  raised  by  the  Moon,  and  in  the 
other  case  later. 

The  Moon  goes  round  the  earth  in  an  elliptical  or- 
bit, and  therefore  in  every  lunar  month  she  approach- 
es nearer  to  the  earth  than  her  mean  distance,  and  re- 
cedes further  from  it.  When  she  is  nearest,  she  at- 
tracts strongest,and  so  raises  the  Tides  most ;  the  con- 
trary happens  when  she  is  farthest,  because  of  her 
weaker  attraction. 


Sec.  10  On  Tides.  123 

When  both-  luminaries  are  in  the  equator,  and  the 
Moon  in  perigee,  (or  at  her  least  distance  from  the 
earth,)  she  raises  the  Tides  highest  of  all ;  especially 
at  her  conjunction  and  opposition,  both  because  the 
equatorial  parts  have  the  greatest  centrifugal  force 
from  their  describing  the  largest  circle,  and  from  the 
concurring  actions  of  the  Sun  and  Moon.  At  the 
change,  the  attractive  forces  of  the  Sun  and  Moon  be- 
ing united,  they  diminish  the  gravity  of  the  waters  un- 
der the  Moon,  and  their  gravity  on  the  opposite  side  is 
diminished  by  means  of  a  greater  centrifugal  force. — 
At  the  full,  while  the  .Moon  raises  the  Tide  under,  and 
opposite  to  her  ;  the  Sun  acting  in  the  same  line,  raises 
the  Tide  under,  and  opposite  to  him ;  whence  their 
conjoint  effect  is  the  same  as  at  the  change,  and  in  both 
cases,  occasion  what  is  called,  Spring  Tides.  But  at  the 
quarters,  the  Sun's  action  diminishes  the  action  of  the 
Moon  on  the  waters,  so  that  they  rise  a  little  under, 
and  opposite  to  the  Sun,  and  full  as  much  under,  and 
opposite  to  the  Moon,  making  what  we  call  Neap  Tides; 
because  the  Sun  and  Moon  then  act  crosswise  to  each 
other.  But  strictly  speaking,  these  Tides  happen  not 
till  some  time  after,  because  in  this,  as  in  other  cases, 
the  actions  do  not  produce  the  greatest  effect,  when 

they  are  at  the  strongest,  but  sometime  afterward. 

The  Sun  being  nearer  the  earth  in  Winter  than  in 
Summer,  is  of  course,  nearer  to  it  in  February  and 
October,  than  in  March  and  September,  and  therefore 
the  greatest  Tides  happen  not  till  some  time  after  the 
autumnal  equinox  :  and  return  a  little  before  the  ver- 


124  On   Tides.  Sec.  10 

nal.  The  Sea  being  thus  put  in  motion,  would  contin- 
ue to  ebb  and  flow  for  several  times,  though  the  Sun 
and  Moon  should  be  annihilated,  or  their  influence 
cease. 

When  the  Moon  is  in  the  equator,  the  Tides  are 
equally  high  in  both  parts  of  the  lunar  day,  or  time  of 
the  Moons'* revolving  from  the  meridian  to  the   meri- 
dian again  ;  which  is  24  hours  and  50  minutes.     But, 
as  the  Moon  declines  from  the  equator  towards  either 
pole,  the  Tides   are   alternately  higher  and  lower  at 
places  having   north  or   south  latitude.     One   of  the 
highest  elevations,  (which  is  that  under  the  Moon,) 
follows  her  towards  the  pole  to  which  she  is  nearest, 
and  the  other  declines  towards  the  opposite  pole  ;  each 
elevation  describing  parallels  as  far  distant  from  the 
equator  on  opposite  sides,  as  the  Moon  declines  from  it 
to  either  side,  and  consequently,  the  parallels  described 
by  these   elevations  of  the  water,  are  twice  as  many 
degrees  from  each  other,  as  the  Moon  is  from  the 
equator ;  increasing  their  distance  as  the  Moon  increas- 
es her  declination',  till  it  be  at  the  greatest ;  when  the 
said  parallels  are  at  a  mean  state  47  degrees  asunder, 
and  on  that^'day,  the  Tides  are  most  unequal  in  their 
heights.     As  the  Moon,  returns  toward  the  equator,the 
parallels  described  by  the  opposite  elevations  approach 
towards  each  other,  until  the  Moon  comes  to  the  equa- 
tor, and  then  they  coincide.     As  the  Moon  declines 
toward  the  opposite  pole  at  equal  distances,  each  ele- 
vation describes  the  same  parallel  in  the  other  part  of 
the  lunar  day,  which  its  opposite  elevations  described 


Sec.  10  On  Tides.  125 

before.  While  the  Moon  has  north  declination,  the 
greatest  Tides  in  the  northern  hemisphere,  are,  when 
she  is  above  the  horizon,  and  the  reverse  when  her  de- 
clination is  south. 

Thus  it  appears,  that  as  the  Tides  are  governed  by 
the  Moon,  they  must  turn  on  the  axis  of  the  Moon's 
orbit,  which  is  inclined  23  degrees  and  28  -minutes  to 
the  earth's  axis  at  a  mean  state,  and  therefore  the  poles 
of  the  Tides,  must  be  so  many  degrees  from  the  poles  of 
the  earth,  or  in  opposite  points  of  the  polar  circles,  go 
ing  around  them  in  every  revolution  of  the  Moon  from 
any  meridian  to  the  same  again. 

It  is  not,  however,  to  be  doubted,  but  that  the  quick 
rotation  of  the  earth  on  his  axis,  brings  the  poles  of 
the  Tides  nearer  to  the  poles  of  the  werld,  than  they 
would  be,  if  the  earth  were  at  rest,  and  the  Moon  re- 
volved about  it.  only  once  a  month,  otherwise  the  Tides 
would  be  more  unequal  in  their  heights,  and  times  of 
their  returns,  than  we  find  they  are.  But  how  near  the 
earth's  rotation  may  bring  the  poles  of  its  axis,  and 
those  of  the  Tides  together,  or  how  far  the  preceding 
Tides  may  effect  those  that  follow,  so  as  to  make  them 
keep  up  nearly  to  the  same  heights  and  times  of  ebbing 
and  flowing,  is  a  problem  more  fit  to  be  solved  by  ob- 
servation than  theory. 

In  open  Seas,  the  Tides  rise  but  to  very  small  heights 
in  proportion  to  what  they  do  in  broad  River?,  whose 
waters  empty  in  the  direction  of  the  stream  of'Ii  Ic  : — 
For,  in  channels  growing  narrower  gradually,  the  wa- 
ter is  accumulated  by  the  opposition  of  the  contracting 


126  On  Tides.  Sec.  10 

bank.  The  Tides  are  so  retarded  in  their  passage 
through  different  shoals  and  channels,  and  otherwise 
so  variously  affected  by  striking  against  Capes  and 
Headlands,  that  to  different  places,  they  happen  at  all 
distances  of  the  Moon  from  the  meridian,  and  conse- 
quently at  all  hours  of  the  lunar  day. 

*  Air  being  lighter  than  water,  and  the  surface  of  the 
atmosphere  nearer  to  the  Moon,  than  the  surface  of  the 
Sea  j  it  cannot  be  doubted  that  the  Moon  raises  much 
higher  Tides  in  the  air,  than  in  the  Sea. 


*  In  a  register  of  the  barometer  kept  for  30  years,  the  Professor  Toal- 
do  of  Padua,  added  together  all  the  heights  of  the  mercury,  when  the  Moon 
was  in  syzigy,  when  she  was  in  quadrature,  and  when  she  was  in  the  apo- 
geal  and  perigeal  points  of  her  orbit.  The  apogeal  exceeded  the  perigeal 
heights  by  14  inches,  and  the  heights  in  syzigy  exceeded  those  in  quadra- 
ture by  11  inches.  The  difference  in  these  heights,  is  sufficiently  great  to 
show  that  the  air  is  accumulated  and  compressed  by  the  attraction  of 
the  Moon. 


Sec.  10  Interrogations  for  Section  Tenth.          127 

Interrogations  for  Section  Tenth. 

By  whom  was  the  cause  of  the  TIDES  first  dis- 
covered ? 

How  does  he  explain  it  ? 

Who  improved  the  idea  of  Kepler  ? 

By  what  does  he  consider  the  waters  to  be  at- 
tracted ? 

Why  are  the  waters  on  the  side  of  the  earth  next  to 
the  Moon,  more  attracted  than  the  central  parts  1 

Why  are  the  central  parts  more  attracted, than  the  wa- 
ters on  the  opposite  side  ? 

From  what  source  is  this  explanation  deduced  ? 

By  what  power  is  the  earth  constantly  falling  towards 
the  Moon,  and  the  Moon  towards  the  earth  1 

If  this  be  actually  the  case,  why  do  they  not  come 
together  ? 

Is  it  the  centre  of  the  earth  that  describes  the  annual 
orbit  round  the  Sun  ? 

Where  is  the  centre  of  gravity  between  the  earth  and 
Moon  ? 

How  much  more  matter  does  the  earth  contain,  than 
the  Moon  1 

What  is  the  centre  of  gravity  between  the  two  bodies  ? 

How  is  it  found  ?  , 

Which  has  the  greatest  influence  in  raising  Tides,the 
Sun  or  Moon  ? 

Are  the  Tides  at  the  highest  when  the  Moon  is  due 
north,  or  south  ? 

What  is  the  reason  ? 


128          Interrogations  for  Section  Tenth.          Sec.  10 

Do  the  Tides  always  answer  to  the  same  distance 
of  the  Moon  from  the  meridian  at  the  same  places  ? 

Does  the  Moon  approach  nearer,  and  recede  farther 
iVom  the  earth  in  each  of  her  revolutions'? 

At  what  time  does  she  attract  the  earth  most? 

At  what  time  does  she  attract  it  the  least  ? 

In  what  position  are  the  Sun  and  Moon  when  the 
highest  Tides  are  raised  1 

What  are  Spring-Tides  ? 

What  are  Neap-Tides? 

In  what  manner  do  the  attractions  of  the  Sun  and 
Moon  act  on  each  other,  to  produce  Spring-Tides  ? 

In  what  manner  to  produce  Neap-Tides  ? 

Where  is  the  moon  when  the  Tides  are  equally  high 
in  both  parts  of  the  lunar  day  ? 

What  is  understood  by  the  lunar  day  ? 

What  is  the  length  of  the  lunar  day  ? 

At  what  time  are  the  Tides  most  unequal  ? 

In  which  hemisphere  are  the  highest  Tides,  when  the 
muon  has  north  declination  ? 

Which  when  in  her  south  declination  ? 

Do  the  Tides  rise  very  high  in  open  Seas  ? 

Are  the  Tides  ever  retarded  in  their  passage  ? 

What  retards  them  ? 

What  are  aerial  Tides  ? 

How  were  they  discovered,  and  by  whom  ? 


SECTION  ELEVENTH. 


PROBLEMS. 

PROBLEM  I. 

To  convert  Time  into  degrees,  minutes,  fyc. 

RULE. 

As  one  hour  is  to  15  degrees,  so  is  the  time  given 
to  the  answer. 

1.  How  many  degrees  are  equal  to  8  hours,  20  min- 

utes, and  30  seconds  1 
2d.  The  Sun  passes  the  meridian  of  Detroit  1  hour,  19 

minutes  after  12  o'clock,  noon  at  Boston,  how  far  are 

those  places  asunder  ? 

PROBLEM  II. 

To  convert  degrees,  minutes,  Sfc.  into  Time. 

RULE. 

As  15  degrees  are  to  an  hour,  so  are  the  number  of 
degrees  given  to  the  time. 

p 


130  .    Astronomical  Problems.  Sec.  11 

EXAMPLES. 

1.  The  apparent  distance  of  Venus  from  the  Sun,  can 
never  be  above  50  degrees,  and  when  at  that  dis- 
tance, how  long  does  she  rise  before  the  Sun,  or  set 
after  him  ? 

2.  The  greatest   elongation  of  Mercury  is   said  to  be 
28  degrees,  20  minutes  and  19   seconds,  how  long 
can  he  set  after  the  Sun,  when  an  evening  star  ? 

PROBLEM  III. 

The  diurnal  arc  of  the  Sun,  or  of  any  planet  being 
given,  to  find  the  time  of  the  rising  or  setting  of  the 
Sun. 

RULE. 

Bring  the  diurnal  arc  into  time  by  Problem  2d. 
Divide  this  time  by  two,  and  the  quotient  will  be  the 
time  at  which  the  Sun  sets.  Take  this  time  from  12 
hours,  and  the  remainder  \vill  be  the  time  at  which  the 
Sun  rises. 

EXAMPLES. 

1.  Suppose  the  Sun's  diurnal  arc  be  174  degrees  and 
thirty  minutes,  at   what  time  does  he  rise  and  set. 
Jlns.  5  hours  49  minutes,  the  time  of  the   Sun's  set- 
ting, and  he  rises  at  6  hours  and  1 1  minutes. 

2.  The  diurnal  arc  of  Venus  is  found  to  be  96  degrees 
and  44  minutes,  at  what  hours  does  the  Sun  rise,and 
when  does  he  set  ? 

3.  The  diurnal  arc  of  Mars,   is  198  degrees,  14  min- 
utes and  50  seconds. 


Sec.  1 1  Astronomical  Problems.  131 

The  diurnal  arc  of  Jupiter,  is  201  degrees,  33  minutes 
and  16  seconds. 

The  diurnal  arc  of  Saturn,  is  1 96  degrees,  and  1 4  min- 
utes :  and  the  diurnal  arc  of  Herschel  is  213  degrees, 
41  minutes,  and  58  seconds  ;  when,  according  to  the 
above  mentioned  numbers,  does  the  Sun  rise  and 
set? 

PROBLEM  IV. 

The  time  which  the  Sun,  or  any  planet  remains 
above  the  horizon  being  given,  to  find  the  length  of  his 
diurnal,  or  nocturnal  arc. 

RULE. 

Divide  the  given  time  by  two,  and  the  quotient  will 
be  the  time  of  the  Sun's  setting.     Take  this  time  from 
12  hours,  and  the  remainder  will  be  the  time  of  his  ri 
sing.     Multiply  the  given  time  by  1 5  degrees,  and  the 
product  will  give  the  Sun's,  or  planet's  diurnal  arc  ;— 
this  subtracted  from  360  degrees,  will  leave  the  noc- 
turnal arc. 

EXAMPLES. 

1.  On  the  fourth  of  July,  the  Sun  rose  at  43  minutes 
past  5  o'clock ;  at  what  time   did  he  set  on  that  day, 
and  what  was  the  length  of  his  diurnal  arc  1 

2.  September  7th,  1825,  the  Sun  rose  at  5  o'clock  and 
52  minutes,  at  what  time  did  he  set,  and  what  are  the 
dimensions  of  both  arcs  ? 


132  Jlstr onomical  Problems.  Sec.  11 

PROBLEM  V. 

, 

To  find  the  time  which  elapses  between  two  conjunc- 
tions, or' two  oppositions,  or  between  one  conjunction, 
and  one  opposition  of  any  two  planets. 

RULE. 

Find  the  difference  between  the  given  daily  motions 
of  the  two  given  planets,  as  given  in  the  following  table 
of  the  daily  motions,  then  say,  as  the  difference  ofttheir 
daily  motions,  is  to  one  day,  so  is  360  degrees,  to  the 
difference  in  the  times  of  the  two  conjunctions,  or  op- 
positions required.  But  for  one  conjunction,  and 
lone  opposition,  or,  for  a  superior  and  an  inferior 
conjunction ;  say  as  the  difference  of  their  daily  mo- 
tions is  to  one  day,  so  is  180  degrees  to  the  time,  which 
elapses  between  a  conjunction,  and  an  opposition  of  the 
two  given  planets. 

TABLE. 

D. 

Mercury's  daily  motion  is  ....     4,0928  degrees. 

Venus',         do.  do.     .....   1,6021 

The  earth's,  do*  do.  .     .     .     .     .     0,9856 

Mars',  do.  do 0,5240 

Jupiter's,      do.  do 0,0831 

Saturn's,       do.  do 0,0335 

Herschel's,  do.  do 0,0118 


Sec.  11  Astronomical  Problems.  133 

fvC.  EXAMPLES. 

1.  How  many  days  elapse  between  a  conjunction,  and 
an  opposition  of  Mercury  and  \enus. 

Thus  Mercury's  daily  motion,  4,0928  degrees 

Less  the  daily  motion  of  Venus,  1,6021—2,4907 
Degrees,then  as  2,4907  d:  1  day  : :  180d:  72,25  days, 
the  time  required. 

2.  How  many  days  is  Venus  a  morning  and  an  evening 
star,  alternately  to  the  earth  ? 

3.  How  many  days  is  Jupiter  a  morning  and  evening 
star,  alternately  to  the  earth  ? 

4.  How  many  days  is  Mercury  east,  and   how  many 
west  of  the  Sun  to  us  ? 

PROBLEM  VI. 

The  heliocentric  longitude  of  any  two  planets  being 
given,  to  find  when  they  will  be  in  heliocentric  con- 
junction. 

RULE. 

Subtract  the  given'  longitude  of  the  planet  nearest 
the  Sun,  from  that  of  the  planet  farthest  from  him,  if 
practicable,  but  if  not,  add'to  the  latter  360  degrees,  and 
then  subtract,  say,  as  the  difference  of  the  daily  motions 
of  the  given  planets  is  to  one  day,  so  is  the  difference  of 
their  longitudes,  to  the  time  when  the  given  planets  will 
be  in  conjunction. 


134  Astronomical  Problems.  Sec.  11 

EXAMPLES. 
1.  At  what  time  were  Mars  and  Venus  in  conjunction, 

after  the  first  of  January,  1823. 

Venus'  daily  motion  is  1,6021—0,5240=1,0781— 
Mars'  longitude  for  January  1st.  1823,  was  311  de- 
grees and  41  minutes :  less  by  285  degrees,  16  minutes, 
the  longitude  of  Venus  at  the  same  time=26  degrees 
and  25  minutes.  Then  as  1 ,0781 :  1  day  : :  26  d.  25  m. 
=24,5,  or  January  25th,  1823. 

TABLE. 

The  Sun's  geocentric  longitude  for  January  1st.  1823, 
was, 

Degrees. 

280—29  win. 

The  heliocentric  longitude  of  Mercu- 
ry, January  1st.  1823,  was  277—25 
That  of  Venus,  was                           285—16 
The  Earth's,  100—20 
That  of  Mars,  311—41 
Jupiter's,  64 — 51 
Saturn's,                                       38 — 56 
Herschel's,                                 277—30 
On  what  day  of  the  year,  1823,  was  Venus  in  con- 
junction with  the  earth  ? 

3.  When  was  Jupiter  in  conjunction  with  the  earth   in 
the  year  1824  ? 

4.  When  were  Venus  and    Jupiter    in    conjunction, 
in  1825  ? 

6.  On  what  day  in  the  year  1832,  did  Jupiter   set ;  at 
the  moment  the  Sun  arose  ? 


Sec.  11  Astronomical  Problems.  135 

PROBLEM  VII. 

When  the  heliocentric  longitude  of  any  planet,  for 
any  given  day  is  known,  to  find  it  for  any  required 
day. 

RULE. 

Find  the  number  of  days  between  the  given,  and  re- 
quired day  :  then  as  one  day  is  to  the  given  planet's 
daily  motion,  so  are  the  days  so  found,  to  the  distance 
which  the  planet  has  revolved  during  that  time.  Add 
this  distance  to  the  planet's  known  longitude,  and  the 
sum,  if  less  than  360  degrees,  will  be  the  longitude  for 
the  required  day,  but  if  more  than  360  degrees,  then 
subtract  360  degrees  from  it,  and  the  remainder  will  be 
the  true  longitude,  & 

EXAMPLES, 


1.  On  the  first  of  January,  1823,  the  heliocentric  longi- 
tude of  Venus  was  285  degrees,  16  min. ;  what  was 
it  on  the  4th  of  July,  in  the  same  year  ?       Jim.  220 
degrees  and  three  minutes. 

2.  On  the  first  of  January,  1823,  the  earth's  longitude 
was  100  degrees  and  20  minutes ,  what  was  its  longi- 
tude on  the  4th  of  July,  1825  ? 

PROBLEM  VIII 

To  determine  whether  Venus  or  Jupiter  will  be  the 
morning  or  evening  star  on  any  given  day. 


136  Astronomical  Problems  Sec.  II 

RULE. 

Find  the  longitude  of  Venus  and  the  longitude  of  the 
earth  for  the  given  day.  If  the  difference  in  longitude, 
counting  from  the  earth's  place  eastward,  be  less  than 
180  degrees,  Venus  will  be  east  of  the  Sun,  and  conse- 
quently evening  star  :  but  if  that  difference  be  greater 
than  180  degrees,  she  will  be  west  of  the  Sun,  and 
therefore  morning  star.  « 

EXAMPLES. 

On  tiie  4th  of  July,  1823,  was  Venus  a  morning  or  an 
evening  star  ? 

The  longitude  of  Venus  on  the  given  day,  will  be 
found  by  Problem  7th,  to  be  220  degrees  and  3  min- 
utes, and  the  earth's  longitude,  for  the  same  day  by  the 
same  Problem,  281  degrees  and  41  minutes  ;  the  dif- 
ference=61  degrees  and  38  minutes;  this  difference  be- 
ing less  than  180  degrees,  shows  that  Venus  is  east 
of  the  Sun,  and  consequently  an  evening  star. 

Did  Jupiter  rise  before,  or  after  the  Sun,  July  4th, 
1832  ? 

How  many  days  in  succession,  can  Venus  be  a  mor- 
ning, or  an  evening  star  ? 

'  '  / 

How  many- days  in  succession,  can  Jupiter  be  a  mor- 
ning or  an  evening  star  ? 


Sec.  11  Astronomical  Problems.  137 

PROBLEM  IX. 

To  determine  the  day  on  which  any  particular  planet 
shall  have  a  given  longitude. 

RULE. 

Subtract  the  longitude  of  the  given  planet  found  in 
the  preceding  table,from  the  given  longitude,if  practic- 
able ;  but  if  the  longitude  of  the  planet  found  in  the  Ta- 
ble, be  greater  than  the  given  longitude,  increase  the 
latter  by  360  degrees,  and  then  subtract;  divide  the 
remainder  by  the  planet's  daily  motion,  as  recorded  in 
the  Table,  and  the  quotient  will  show  the  number  of 
days  from  the  first  of  January,  when  the  planet  will 
have  the  given  longitude. 

EXAMPLES. 

1.  On  what  day  of  the  year  1823,  did  Venus  have  220 
degrees  of  heliocentric  longitude  ?    Answer — fourth 
of  July. 

2.  On  what  days  in  the  year  1825,  did  each   of  the 
planets  enter  Virgo  ? 

PROBLEM  X. 

To  find  whether  Venus  or  Mercury  will  cross  the 
Sun's  disk  in  any  given  year. 

RULE. 

Find  by  Problem  9th  when  Venus  will  pass  her  node. 
Find  the  earth's  heliocentric  longitude  for  that  day,  and 
if  it  equals  the  longitude  of  Venus'  node,  there  will  be 
a  transit  of  Venus,  and  in  no  other  case.  The  same  may 
be  said  of  the  planet  Mercury. 

Q 


138  Astronomical  Problems.  Sec.  1 1 

EXAMPLES. 

Were  there  a  transit  of  Venus  in  the  year  1824,  or 
not  ? 

The  longitude  of  the  ascending  node  of  Venus,  is  75 
degrees  and  8  minutes,  which  she  passed  on  the  26th 
of  June.  The  earth's  longitude  on  that  day,  was  274 
degrees  and  44  minutes.  The  longitude  of  the  descen- 
ding node  of  Venus,  was  258  degrees  and  8  minutes, 
which  she  passed  on  the  5th  of  March.  The  earth's 
longitude  on  that  day,  was  164  degrees  and  55  minutes, 
consequently  there  was  no  transit  of  Venus  in  1824. 

PEOBLEM  XL 

To  find  when  any  two  given  planets  shall  have  a 
given  heliocentric  aspect,  taking  their  longitudes  as  sta- 
ted in  the  Table  for  1823. 

RULE. 

Add  the  degrees  in  the  aspect  given  to  the  heliocen- 
tric longitude  of  either  given  planet.  Find  the  differ- 
ence between  that  sum  and  the  heliocentric  longitude 
of  the  other  given  planet  :  Then  say,  as jthe  difference 
in  the  daily  motions  of  the  two  given  planets,  is  to  one 
day,so  is  the  difference  in  their  longitude  found  as  above 
to  the  answer  required., 

EXAMPLES. 

At  what  time  in  the  year  1824,  did  the  earth  and  Ve- 
nus have  a  trine  aspect  ? 


Sec.  1 1  Astronomical  Problems.  139 

The  longitude  of  the  earth  for  January  1st.  for  that 
3  ear,  was  100  degrees  and  6  minutes,  to  the  earth's  lon- 
gitude, add  120  degrees,  (the  given  aspect,)  and  the 
sum  is  220  degrees  and  6  minutes. 

The  longitude  of  Venus  on  the  first  day  of  January, 
1824,  was  150  degrees  and  two  minutes  ;  the  difference 
was  70  degrees  and  4  minutes  5  Then  1,6021  degrees — 
,9S56=,6165  difference  of  daily  motion.  Then  ,6165  : 
1  day  :  :  70  d.  4  minutes  :  113  days,  or  the  22d.  of  April. 

2.  On  what  day  were  the  earth  and  Jupiter  in  conjunc- 
tion in  the  year  1826  ? 

3.  When  in  1835,  will  the  earth  and  Venus  be  in  con- 
junction 1 


NOTE.— The  preceding  PROBLEMS  would  be  correct,  if  the 
Planets  moved  in  perfect  circular  orbits,  which  however  is  not  the  fact, 
yet  they  approach  so  near  to  circles,  that  deductions  founded  upon  their 
figure  as  circles,  are  sufficiently  accurate  for  ordinary  calculations. 


SECTION  TWELFTH. 


ON  ECLIPSES. 

IN  the  Solar  System,  the  Sun  is  the  great  fountain 
of  Light,  and  every  planet  and  satellite  is  illuminated 
by  him,  receive  the  distribution  of  his  rays,  and  are  ir- 
radiated by  his  beams.  The  rays  of  light  are  seen  in 
direct  lines,  and  consequently  are  frequently  intercept- 
ed by  the  dark  and  opaque  body  of  the  Moon,  passing 
directly  between  the  earth  and  the  Sun  ;  and  hiding  a 
portion,  or  the  whole  of  his  disk  from  the  view  of  those 
parts  of  the  earth  where  the  penumbra,  or  the  shadow 
of  the  Moon  happens  to  fall.  This  is  called  an  ECLIPSE 
OF  THE  SUN. 

It  is  only  at  the  time  of  new  Moon,  that  an  Eclipse 
of  this  kind  can  possibly  take  place,  and  then  only  when 
the  Sun  is  within  seventeen  degrees  of  either  the  as- 
cending or  descending  nodes ;  for  if  his  distance  at  the 
time  of  new  Moon  be  greater  than  seventeen  degrees 
from  either  node,  no  part  of  the  Moon's  shadow  will 
touch  the  earth,  and  consequently  there  will  be 
no  Eclipse. 

The  orbit  in  which  the  Moon  really  moves,  is  differ- 
ent from  the  ecliptic,  one  half  being  elevated  five  and 


Sec.  12  On  Eclipses.  141 

one -third  degrees  above  it,  and  the  other  half  as  much 
depressed  below.  The  Moon's  orbit  therefore  inter- 
sects the  ecliptic  in  two  points  diametrically  opposite 
to  each  other,  and  these  intersections  are  called  the 
Moon's  nodes.  The  Moon,  therefore  can  never  be  in 
the  ecliptic,  but  when  she  is  in  either  of  her  nodes ; 
which  is  at  least  twice  in  every  lunation,  or  course 
from  change  to  change,  and  sometimes  thrice.  That 
node  from  which  the  Moon  begins  to  ascend  northward, 
or  above  the  ecliptic  in  northern  latitudes  is  called  the 
ascending  node  ;  and  the  other  the  descending  node  ; 
because  the  Moon  when  she  passes  by  it  descends 
below  the  ECLIPTIC  southward.  The  ECLIPTIC 
is  the  great  circle  which  the  earth  describes 
in  its  annual  revolution  around  the  Sun,  and  is 
divided  into  twelve  equal  parts,  of  thirty  degrees  each 
called  signs.  Six  of  these,  namely,  Aries,  Taurus, 
Gemini,  Cancer,  Leo  and  Virgo  are  north  ;  and  the 
other  six,  to  wit,  Libra,  Scorpio,  Sagitarius,  Capricor- 
nus,  Aquarius  and  Pisces,  south  of  the  equotor.* 

When  the  earth  comes  between  the  Sun  and  Moon, 
the  Moon  passes  through  the  earth's  shadow,  and  hav- 
ing no  light  of  her  own,  she  suffers  a  real  Eclipse ;  the 
rays  of  the  Sun  being  intercepted  by  the  earth.  This 
can  only  happen  at  the  time  of  full  Moon ;  and,  when 
the  Sun  is  within  twelve  degrees  of  the  Moon's  ascen- 
ding or  descending  nodes.  Should  the  Sun's  distance 


*  The  Equator  is  an  imaginary  circle  passing  round  the  earth  from 
east  to  west,  dividing  it  into  two  equal  parts,  called  Hemispheres. 


142  On  Eclipses.  Sec.  12 

from  the  node  exceed  twelve  degrees,  the  shadow  of 
the  earth  would  no  where  touch  the  surface  of  the 
Moon,  and  consequently  she  could  not  suffer  an  Eclipse. 

When  the  Sun  is  Eclipsed  to  us,  the  inhabitants  of 
the  Moon  on  the  side  next  the  earth,  see  her  shadow 
like  a  dark  spot  travelling  over  the  earth  about  twice 
as  fast  as  its  equatorial  parts  move,  and  the  same  way. 

When  the  earth  passes  between  the  Sun  and  Moon, 
the  Sun  appears  in  every  part  of  the  Moon  where  the 
earth's  shadow  falls  totally  Eclipsed  ;  and  the  duration 
is  as  long  as  she  remains  in  the  earth's  shadow. 

If  the  earth  and  Sun  were  of  equal  sizes,  the  shadow 
of  the  earth  would  be  infinitely  extended,  and  wholly 
of  the  same  breadth,  and  the  planet  Mars  when  in  ei- 
ther of  her  nodes,  and  in  opposition  to  the  Sun,  (al- 
though forty-two  millions  of  miles  from  the  earth,) 
would  be  Eclipsed  by  the  shadow.  If  the  earth  were 
larger  than  the  Sun,  her  shadow  would  be  sufficient  to 
Eclipse  the  larger  planets,  Jupiter  and  Saturn  with  all 
their  satellites,  when  they  were  opposite  to  him ;  but 
the  shadow  of  the  earth  terminates  in  a  point  long  be- 
fore it  reaches  any  of  the  primary  planets.  It  is  there- 
fore evident,  that  the  earth  is  much  less  than  the  Sun, 
or  its  shadow  could  not  end  in  a  point  at  so  short  a  dis- 
tance. 

If  the  Sun  and  Moon  were  of  equal  sizes,  she  would 
cast  a  shadow  on  the  earth's  surface  of  more  than  two 
thousand  miles  in  breadth,  even  if  it  fell  directly  against 
its  centre.  But  the  shadow  of  the  Moon  is  seldom 
more  than  one  hundred  and  fifty  miles  in  breadth  at 


.  12  On  Eclipses.  143 

the  earth,  unless  in  total  Eclipses  of  the  Sun,  her  shad- 
ow strikes  on  the  earth  in  a  very  oblique  direction. 

In  annular  Eclipses,  the  Moon's  shadow  terminates 
in  a  point  at  some  distance  before  it  reaches  the  earth  ; 
and  consequently  the  Moon  is  mnch  less  than  the  Sun. 
If  the  Moon  were  actually  thrice  its  present  size,  it 
would  still  in  many  instances,  be  totally  Eclipsed.  A 
sufficient  proof  of  this,  is  given  by  her  long  continuance 
in  the  earth's  shadow,  during  any  of  her  total  Eclipses. 
Therefore  the  diameter  of  the  earth  is  more  than  three 
times  the  diameter  of  the  Moon. 

Though  all  opaque  bodies,  on  which  the  Sun  shines, 
have  their  shadows  ;  yet  such  is  the  magnitude  of  the 
Sun,  and  the  distances  of  the  planets,  that  the  prima- 
ries can  never  Eclipse  each  other.  A  primary  can  only 
Eclipse  its  secondary,  or  be  Eclipsed  by  it,  and  never 
by  those  except  when  they  are  in  opposition 
or  conjunction  with  the  Sun,  as  before  stated.  The 
primary  planets  are  very  seldom  in  such  positions,  but 
the  Sun  and  Moon  are,  in  every  month. 

If  the  Moon's  orbit  were  coincident  with  the  plane 
of  the  ecliptic,  in  which  the  earth  wheels  its  stated 
courses,  the  Moon's  shadow  would  fall  on  the  earth  at 
every  change,  and  the  Sun  be  eclipsed  to  every  part 
of  the  earth  where  the  penumbra  happened  to  fall.  In 
the  same  manner,  the  Moon  would  have  to  travel 
through  the  middle  of  the  earth's  shadow,  and  be  to- 
tally Eclipsed  at  every  full.  The  duration  of  total 
darkness  in  every  instance,  exceeding  an  hour  and  a 
half. 


%; 

144  On  Eclipses.  Sec.  12 

A  question  like  the  following  naturally  arises  : 

Why  is  it  that  the  Sun  is  not  Eclipsed  at  every  change, 
if  the  Moon  actually  passes  between  the  Sun  and  the 
earth.  And  why  is  not  the  Moon  Eclipsed  at  every 
full,  if  the  earth  passes  between  the  Sun  and  Moon  in 
every  month  1 

One  half  of  the  Moon's  orbit,  is  elevated  5  degrees 
and  twenty  minutes  above  the  ecliptic,  and  the  other 
half  is.  as  much  depressed  below  it ;  and,  as  before  has 
been  observed,  the  Moon's  orbit  intersects  the 
ecliptic,  in  two  opposite  points,  called  the  MOON'S 
NODES. 

When  these  points  are  in  a  right  line  with  the  cen- 
tre of  the  Sun  at  new  or  full  Moon,  the  Sun,  Moon,and 
earth  are  all  in  a  right  line  ;  and  if  the  Moon  be  then 
new;  her  shadow  falls  upon  the  earth,  but  if  she  be  full, 
the  earth's  shadow  falls  upon  her.  When  the  Sun  and 
Moon  are  more  than  1 7  deg's.  from  either  of  the  nodes 
at  the  time  of  conjunction,the  moon  is  generally  too  high 
or  too  low  in  her  orbit  to  cast  any  part  of  her  shadow 
on  the  surface  of  the  earth.  And  when  the  Sun  is  more 
than  12  degrees  from  either  of  the  nodes  at  the  time 
of  full  Moon,  the  Moon  is  generally  either  two  high  or 
too  low  to  pass  through  any  part  of  the  earth's  shadow ; 
therefore  in  both  these  cases  there  can  be  no  Eclipse. 

This  howevever  admits  of  some  variation,  for  in 
apogeal  Eclipses,  the  solar  limit  is  only  sixteen  de- 
grees and  thirty  minutes,  and  in  perigeal  it  is  eighteen 
degrees  and  twenty  minutes.  When  the  full  Moon  is 


ee:  12  On  Eclipses.  145 

in  her  apogee,  *  she  will  be  Eclipsed,  if  she  be  within 
16  degrees  and  thirty  minutes  of  the  node  ;  and  when 
in  her  perigee,  if  within  twelve  degrees  and  two 
minutes. 

The  moon's  orbit  contains  360  deg's.  of  which  the  lim- 
its of  seventeen  degrees  at  a  mean  rate  for  Solar  Eclip- 
ses, fy  twelve  for  lunar  are  only  small  portions,  and  the 
Sun  generally  passes  by  the  nodes  only  twice  in  a  yean 
and  consequently  impossible  that  Eclipses  should  hap- 
pen in  every  month.  If  the  line  of  the  nodes,  like  the 
axis  of  the  earth,  were  carried  parallel  to  itself  around 
the  Sun,  there  would  be  exactly  half  a  year  between 
the  conjunctions  of  the  Sun  and  nodes.  But  the  nodes 
shift  backward,  or  contrary  to  the  earth's  annual  mo- 
tion, nineteen  degrees  and  twenty  minutes  every  year ; 
and  therefore  the  same  node  comes  round  to  the  Sun 
nineteen  days  sooner  every  year,  than  in  the  one  pre- 
ceeding.  173  days,  therefore,  after  the  ascending 
node  has  passed  by  the  Sun,  the  descending  node 
also  passes  by  him.  In  whatever  season  of  the  year 
the  luminaries  are  Eclipsed,  in  1 73  days  after,  we  may 
expect  Eclipses  about  the  opposite  node.  The  nodes 
shift  through  all  the  signs  and  degrees  of  the  ecliptic 
in  18  years  and  225  days,  in  which  time  there  would 
always  be  a  regular  periodical  return  of  Eclipses,  if 
any  number  of  lunations  were  completed  without  a 


*  The  fartherest  point  of  each  orbit  from  the  earth's  centre  is  called 
the  apogee,  and  the  nearest  point  is  called  the  perige"e.  These  points  are 
directly  opposite  each  other,  and  consequently  exactly  six  signs  asunder. 

R 


146  On  Eclipses.  Sec.  12 

fraction.  But  this  never  happens,  for  if  both  the  Sun 
and  Moon  should  start  from  a  line  of  conjunction  with 
either  of  the  nodes  in  any  point  of  the  ecliptic,  the  Sun 
would  perform  18  annual  revolutions  and  222  degrees 
of  the  19th,  and  the  Moon  230  lunations,  and  85  de- 
grees of  another  by  the  time  the  node  came  around  to 
the  same  point  of  the  ecliptic  again. 

The  Sun  would  then  be  138  degrees  from  the  node, 
and  the  Moon  85  degrees  from  the  Sun.  In  223  mean 
lunations  after  the  Sun,  Moon  and  node,  have  been  in 
a  line  of  conjunction,  they  return  so  nearly  to  the  same 
state  again,  that  the  same  node  which  was  in  conjunc- 
tion with  the  Sun  and  Moon  at  the  commencement  of 
these  lunations,  will  be  within  28  minutes,  and  12  sec- 
onds df  a  degree  of  a  line  of  conjunction  with  the  Sun 
and  Moon  again,  when  the  last  of  these  lunations  is 
completed.  In  that  time,  there  will  be  a  regular  pe- 
riod of  Eclipses,  or  rather  a  periodical  return  of  the 
same  eclipse  for  many  ages.  In  this  period,  (which 
was  first  discovered  by  the  Chaldeans,)  there  are  18 
Julian  years,  1 1  days,  7  hours,  43  minutes,  and  21 
seconds,  when  the  29th  day  of  February  in  leap  years, 
is  four  times  included  ;  but  one  day  less  when  included 
5  times.  Consequently,  if  to  the  mean  time  of  any 
Eclipse,  whether  of  the  Sun  or  Moon,  the  above  named 
time  be  added,  you  will  have  the  mean  time  of  its  pe- 
riodical return.  But  the  falling  back  of  the  line  of  con- 
junctions, or  oppositions  of  the  Sun  and  Moon,  namely, 
28  minutes,  12  seconds,  with  respect  to  the  line  of  the 
nodes  in  every  period,  will  wear  it  out  in  process  of 


12  On  Eclipses.  147 

time,  so  that  the  shadow  will  not  again  touch  the  earth 
or  Moon,  during  the  space  of  12,492  years.  Those 
Eclipses  of  the  Sun  which  happen  about  the  ascending 
node,  and  begin  to  come  in  at  the  north  pole  of  the 
earth,  will  continue  at  each  periodical  return  to  ad- 
vance southwardly,  until  they  leave  the  earth  at  the 
south  pole;  and  the  contrary  with  those  that  happen 
about  the  descending  node,  and  come  in  at  the  south 
pole.  From  the  time  that  an  Eclipse  of  the  Sun  first 
touches  the  earth,  until  it  completes  its  periodical  re- 
turns, and  leaves  the  same,  there  will  be  77  periods 
equal  to  1388  years.  The  same  Eclipse  cannot  then 
again  touch  this  earth,  in  a  less  space  than  12492  years 
as  above  stated. 

If  the  motions  of  the  Sun,  Moon,  and  nodes  were  the 
same  in  every  part  of  their  orbits,  we  should  need  no- 
thing more  than  what  has  been  said  to  find  the  exact 
time  of  all  Eclipses ;  but  as  this  is  not  the  case,  we  are 
under  the  necessity  of  forming  Tables  so  constructed, 
that  the  mean  time  can  be  reduced  to  the  true.  By 
the  following  example,  it  will  be  found,  that  by  the  true 
motions  of  the  Sun,  Moon  and  nodes,  the  Eclipse  cal- 
culated, leaves  the  earth  five  periods  sooner  than  it 
would  have  done,  by  mean  equable  motions.  To  ex- 
emplify this  matter  more  fully,  I  will  take  the  Eclipse 
of  the  Sun,  which  happened  in  the  year  1764,  March 
21st.  Old  Style,  (or  April  1st.  in  the  new,)  according  to 
its  mean  revolutions,  and  also  according  to  its  true 
equated  time. 


m 

148  On  Eclipses.  Sec.  12 

The  shadow,  or  penumbra  of  the  Moon,  fell  in  open 
space  at  each  return,  without  touching  the  earth  ever 
since  the  creation,  until  the  year  of  our  Lord,  1 295  ; 
then  on  the  1 4th  day   of  June,   at  52  minutes,  and  59 
seconds  in  the  morning,  Old  Style,  the  Moon's  shadow 
touched  the  earth   at  the  north   pole.     In   each  suc- 
ceeding period  since  that  time,  the  Sun  has  come  28 
minutes  and  12  seconds  nearer  the  same  node,  and  the 
Moon's  shadow  has  gone  more  southwardly.      In  the 
year  1962,  on  the  18th  of  July,  Old  Style,   (or  31st.  in 
the  new,)    at  10  hours.  36  minutes,  21  seconds  in  the 
afternoon,  the   same   Eclipse   will  have   returned  38 
times.     The  Sun  will  then  be  only  24  minutes  and  45 
seconds  from  the  ascending  node,  and  the  centre  of  the 
Moon's  shadow  will  fall  a  little  north  of  the  equator.— 
At  the  end  of  the  next  following  period,  in  the  year 
1980,  July  29th,  Old  Style,  (or  August  llth  in  the 
new,)  at  6  hours,  19  minutes  and  41  seconds  in  the 
morning,  the  Sun  will   have  receded  back  three  min- 
utes and  twenty-seven   seconds   from  the  ascending 
node  ;  the  Moon  will  then   have  a  small  degree  of 
south  latitude,  and  consequently  cast  her  shadow  a  lit- 
tle south  of  the  equator.     After  this,  at  every  follow- 
ing period,  the  Sun  will  be  28  minutes  and  12  seconds 
further  back  from  the  ascending  node  than  at  the  pre- 
ceding, and  the  Moon's  shadow  will  continue  at  each 
succeeding  period  to  approach  nearer  the  south  pole, 
until  September  13,  Old  Style,  (or  October  1st.  in  the 
new,)  at  1 1   hours,  46  minutes  and  22  seconds  in  the 
morning,  in  the  year  2665,  when  the  Eclinse  will  have 


Sec.  12  On  Eclipses.  149 

completed  its  77th  periodical  return,  and  the  shadow 
of  the  Moon  leaves  the  earth  at  the  south  pole  to  re- 
turn no  more,  until  the  lapse  of  12492  years.  But  on 
account  of  the  true,  (or  unequable)  motions  of  the  Sun, 
Moon,  fy  nodes,  the  first  coming  m  of  this  Eclipse  at  the 
north  pole  of  the  earth,  was  on  the  24th  of  June,  1313, 
at  3  hours,  57  minutes,  and  3  seconds  in  the  afternoon, 
and  it  will  finally  leave  the  earth  at  the  south  pjde  on 
the  18th  day  of  August,  (according  to  New  Style,)  in 
the  year  2593,  at  10  hours,  25  minutes  and  31  seconds 
afternoon,  at  the  72d.  period.  So  that  the  true  motions 
do  not  only  alter  the  true  times  from  the  mean,  but 
they  also  cut  off,  five  periods  from  those  of  the  mean 
returns  of  this  Eclipse. 

In  any  year,  the  number  of  Eclipses  of  both  lumina- 
ries cannot  be  less  than  two,  nor  more  than  seven  ;  the 
most  usual  number  is  four,  and  it  is  very  rare  to  have 
more  than  six.  The  Eclipses  of  the  Sun  are  more 
frequent  than  those  of  the  Moon,  because  the  Sun's 
ecliptic  limits  are  greater  than  those  of  the  Moon's. — 
(The  proportions  being  as  17  is  to  12,)  yet  we  have 
more  visible  Eclipses  of  the  Moon,  than  of  the  Sun  ; 
because  Eclipses  of  the  Moon  are  seen  from  all  parts 
of  that  hemisphere  of  the  earth  which  is  next  her  ;  and 
are  equally  great  to  each  of  those  parts  ;  but  Eclipses 
of  the  Sun  are  only  visible  to  that  small  portion  of  the 
hemisphere  next  him,  whereon  the  Moon's  shadow 
happens  to  fall. 

The  Moon's  orbit  being  elliptical,  and  the  earth  in 
one  of  its  focuses ;  she  is  once  at  her  least  distance 


150  On  Eclipses.  Sec.  12 

from  the  earth,  and  once  at  her  greatest  in  every  lu- 
nation or  revolution  around  the  earth.  When  the 
Moon  changes  at  her  least  distance  from  the  earth,  and 
so  near  the  node  that  her  dark  shadow  falls  on  the 
earth ;  she  appears  sufficiently  large  to  cover  the 
whole  disk  of  the  Sun  from  that  part  on  which  her 
shadow  falls,  and  the  Sun  appears  totally  eclipsed  for 
the  space  of  four  minutes. 

But  when  she  changes  at  her  greatest  distance  from 
the  earth,  and  so  near  the  node  that  her  dark  shadow 
is  directed  towards  the  earth,  her  diameter  subtends  a 
less  angle  than  the  Sun's,  and  therefore  cannot  hide  the 
whole  disk  from  any  part  of  the  earth,  nor  does  her 
shadow  reach  it  at  that  time ;  and  to  the  place  over 
which  the  point  of  her  shadow  hangs,  the  Eclipse  is 
annular,  and  the  edge  of  the  Sun  appears  like  a  lumin- 
ous ring  around  the  whole  body  of  the  Moon.  When 
the  change  happens  within  17  degrees  of  the  node,  and 
the  Moon  at  her  mean  distance  from  the  earth,  the 
point  of  her  shadow  just  touches  the  earth,  and  the 
Sun  is  totally  eclipsed  to  that  small  spot  on  which  the 
Moon's  shadow  falls  ;  but  the  duration  of  total  dark- 
ness is  not  of  a  moment's  continuance.  The  Moon's 
apparent  diameter  when  largest,  exceeds  the  Sun's 
when  least,  according  to  the  calculations  of  modern 
Astronomers,  two  minutes  and  five  seconds,  the  dura- 
tion of  total  darkness,  therefore  may  at  such  time  con- 
tinue four  minutes  and  six  seconds ;  casting  a  shadow 
on  the  earth's  surface  of  180  miles  broad.  When  the 
Moon  changes  exactly  in  the  node,  the  penumbra  is 


See.  12  On  Eclipses.  151 

circular  on  the  earth  at  the  middle  of  the  general 
Eclipse,  because  at  that  time  it  falls  perpendicularly  on 
the  earth's  surface ;  but  in  every  other  moment,  it  falls 
obliquely,  and  therefore  will  be  elliptical,  and  the  more 
so,  as  the  time  is  longer  after  the  middle  of  the  gen- 
eral Eclipse  ;  and  then  much  greater  portions  of  the 
earth  are  involved  in  the  penumbra. 

When  the  penumbra  first  touches  the  earth  the  gen- 
eral Eclipse  begins,  and  it  ends  when  it  leaves  the 
earth :  from  the  beginning  to  the  end,  the  sun  appears. 
Eclipsed  in  some  part  of  the  earth  or  other.,  When 
the  penumbra  touches  any  place,  the  Eclipse  «begins 
at  that  place,  and  ends,  when  the  penumbra  leaves  it. 
When  the  moon  changes  exactly  in  the  node  the  pe- 
numbra goes  over  the  centre  of  the  earth  as  seen  from 
the  moon,  and  consequently  by  describing  the  longest 
line  possible  on  the  earth  continues  the  longest  upon  it ; 
namely  at  a  mean  rate  five  hours  and  fifty  minutes  : 
more,  if  the  moon  be  at  her  greatest  distance  from 
the  earth,  because  she  then  moves  slowest,  and  less  if 
she  be  at  her  nearest  approach,  because  of  her  ac- 
celerated motion. 

The  moon  changes  at  all  hours,  and  as  often  in  one 
node  as  in  the  other,  and  at  all  distances  from  them 
both,  at  different  times  as  it  happens ;  the  variety  of 
phases  of  Eclipses  are  therefore  almost  innumerable, 
even  at  the  same  places,  considering  also  how  various- 
ly the  same  places  are  situated  on  the  enlightened  disk 
of  the  earth  with  respect  to  the  motion  of  the  penum- 
bra, at  the  different  hours  when  Eclipses  happen. 


152  On  Eclipses.  Sec.  12 

When  the  Moon  changes  17  degrees  short  of  her 
descending  node,  the  penumbra  just  touches  the  north- 
ern part  of  the  earth's  disk  near  the  north  pole,  and 
as  seen  from  that  place,  the  Moon  appears  to  touch 
the  Sun,  but  hides  no  part  of  him  from  sight.  Had  the 
change  been  as  far  short  of  the  ascending  node,  the 
penumbra  would  have  touched  the  southern  part  of  the 
disk  near  the  south  pole.  When  the  Moon  changes 
12  degrees  short  of  the  descending  node,  more  than  a 
third  part  of  the  penumbra,  falls  on  the  northern  parts 
of  the  earth  at  the  middle  of  the  general  Eclipse.  Had 
she  changed  as  far  past  the  same  node,  as  much  of  the 
other  side  of  the  penumbra  would  have  fallen  on  the 
southern  parts  of  the  earth,  all  the  rest  in  open  space. 

When  the  Moon  changes  6  degrees  from  the  node, 
almost  the  whole  penumbra  falls  on  the  earth  at  the 
the  middle  of  the  general  Eclipse. 

The  further  the  Moon  changes  from  either  node 
within  17  degrees  of  it,  the  shorter  is  the  penumbra's 
continuance  on  the  earth  ;  because  it  goes  over  a  less 
portion  of  the  disk.  The  nearer  the  penumbra's  cen- 
tre is  to  the  equator  at  the  middle  of  the  general 
Eclipse,  the  longer  is  its  duration  at  places  where  it  is 
central  ;  because  the  nearer  that  any  place  is  to  the 
equator,  the  greater  is  the  circle  it  describes  by  the 
earth's  motion  on  its  axis,  and  the  place  moving  quick 
keeps  longer  in  the  penumbra,  whose  motion  is  the 
same  way  with  that  of  the  place,  though  faster  as  has 
been  mentioned.  That  Eclipses  of  the  Moon  can  never 
happen  only  at  the  time  of  full,  and  the  reason  why  she 


Sec.  12  On  Eclipses.  153 

is  not  eclipsed  at  every  full,  has  already  been  men- 
tioned. 

The  Moon  when  totally  eclipsed,  (though  a  dark 
opaque  body,  and  shines  only  by  reflection,)  is  not  in- 
visible, if  she  be  above  the  horizon,  and  the  sky  clear ; 
but  generally  appears  of  a  dusky  color  which  some 
have  thought  to  be  her  native  light.  But  the  true  cause 
of  her  being  visible,  is  the  scattered  beams  of  the  Sun, 
bent  into  the  earth's  shadow  by  going  through  the  at- 
mosphere, which  being  more  dense  near  the  earth,  than 
at  considerable  heights  above  it,  refracts,  or  bends  the 
rays  of  the  Sun  more  inward  the  nearer  they  are  pass- 
ing by  the  earth's  surface,  than  those  rays  which  go 
through  higher  parts  of  the  atmosphere  where  it  is  less 
dense;  according  to  its  height,  until  it  be  so  thin,  or 
rare  as  to  lose  its  refractive  power. 

When  the  MocJn  goes  through  the  centre  of  the  earth's 
shadow,  she  is  directly  opposite  to  the  Sun,  yet  the 
Moon  has  been  often  seen  totally  eclipsed  in  the  hori- 
zon, when  the  Sun  was  also  visible  in  the  opposite  part 
of  it ;  for  the  horizontal  refraction  being  almost  34 
minutes  of  a  degree,  and  the  diameter  of  the  Sun  and 
Moon  being  each  at  a  mean  state  but  32  minutes,  the 
refraction  causes  both  luminaries  to  appear  above  the 
horizon,  when  they  are  actually  below  it.  When  the 
Moon  is  full  at  12  degrees  from  either  node,  she  just 
touches  the  earth's  shadow,  but  does  not  enter  into  it. 
When  she  is  full  at  6  degrees  from  either  node,  she  is 
totally,  but  not  centrally  immersed  in  the  earth's  shad- 
ow, she  takes  the  longest  line  possible,  which  is  the 


154  On  Edipses.  Sec.  12 

diameter  through  it,  and  such  an  Eclipse,  (being  both 
total,  and  central,)  is  of  the  longest  duration,  namely, 
three  hours,  57  minutes  and  6  seconds  from  the  begin- 
ning to  the  end,  if  the  Moon  be  at  her  greatest  distance 
from  the  earth  ;  and  3  hours,  37  minutes  and  26  sec- 
onds, if  she  be  at  her  least  distance. 

The  reason  of  this  difference  is,  that  when  the  Moon 
is  farthest  from  the  earth,  her  motions  are  retarded,  but 
when  nearest  to  the  earth,  her  motions  are  accelerated. 


Sec.  12          Interrogations  for  Section  Tkctlftk.          165 


. 

«  X 

Interrogotions  for  Section  Twelfth. 

Arc  the  rays  of  light  proceeding  from  the  Sun,  fre- 
quently intercepted  ? 

By  what  are  they  intercepted  ? 
What  is  understood  by  the  penumbra  ? 
What  is  an  Eclipse  of  the  Sun  ? 
At  what  stage  of  the  Moon  does  an  Eclipse  of  the 
Sun  happen  1 

How  near  to  either  of  the  nodes  must  the  Sun  be  to 
suffer  an  Eclipse  1 

Does  the  Moon's  orbit  differ  from  the  ecliptic  ? 

What  is  the  ecliptic  7 

What  are  the  Moon's  nodes  1 

Why  cannot  the  Sun  be  eclipsed  unless  he  be  within ; 
17  degrees  of  the  node  ? 

How  often  is  the  Moon  in  the  ecliptic  ? 
Which  is  called  the  ascending  node  ? 
Which  is  called  the  descending  node  1 
What  is  an  Eclipse  of  the  Moon  ? 
At  what  stage  of  the  Moon  does  this  happen  ? 

How  near  must  the  Sun  be  to  either  of  the   node.?,  so 
that  the  Moon  can  suffer  an  Eclipse  ? 
What  causes  it  ? 


156         Interrogations  for  Section  Twelfth.         Sec.  12 

Should  the  same  distance  from  either  node  at  the 
time  of  full  Moon  exceed  twelve  degrees,  could  the 
shadow  of  the  earth  touch  the  surface  of  the  Moon  ? 

As  the  Moon  passes  between  the  Sun  and  the  earth 
at  every  new  Moon,  why  is  not  the  Sun  eclipsed  at  eve- 
ry new  Moon  ? 

Why  is  not  the  Moon  eclipsed  at  every  full  ? 

What  is  the  farthest  point  of  each  orbit  from  the 
earth's  centre  called  1 

What  the  nearest  point  ? 

How  many  times  a  year  does  the  Sun  generally  pass 
by  the  nodes  ? 

In  what  time  do  the  nodes  pass  through  all  the  signs 
of  the  ecliptic  ? 

How  many  lunations  after  the  Sun,  Moon  and  nodes 
have  been  in  conjunction,  before  they  return  nearly  to 
the  same  state  again  1 

What  is  a  periodical  return  of  an  Eclipse  ? 
Are  the  motions  of  the  Sun,  Moon  and  nodes  the 
same  in  every  part  of  their  orbits  ? 

How  can  the  mean  time  of  these  conjunctions  be  re- 
duced to  the  true  ? 

How  many  are  the  greatest  number  of  Eclipses  that 
can  possibly  happen  in  one  year  ? 

How  many  the  least  ? 

What  the  most  usual  number  ? 


Sec.  12          Interrogations  for  Section  Twelfth.          157 

Which  is  the  most  frequent,  those  of  the  Sun  or 
Moon  ? 

What  is  the  reason  1 

Are  there  more  visible  Eclipses  of  the  Moon  than  of 
the  Sun  ? 

What  is  the  reason  1 

What  is  a  total  Eclipse  of  the  Sun  ? 

How  long  can  the  Moon  hide  the  whole  face  of  the 
Sun  from  our  view  ? 

In  what  part  of  her  orbit  must  the  Moon  be  to  cause 
a  total  Eclipse  ? 

What  is  an  annular  Eclipse  ? 

How  many  miles  in  diameter  would  the  shadow  of 
the  Moon  be  on  the  earth,  in  an  Eclipse  when  total 
darkness  continues  four  minutes  ? 

When  the  Moon  changes  exactly  in  the  node,  what  is 
the  form  of  the  shadow,  and  where  does  it  strike  the 
earth  ? 

When  does  an  Eclipse  begin  ? 

When  does  it  end  1 

When  does  it  begin  and  end  at  any  particular  place  1 
When  the  Moon  changes  17  degrees  short  of  her  de- 
scending   node,    where    will    her    shadow  touch  the 
earth  ? 

If  as  far  short  of  her  ascending  node,  where  on  the 
earth  will  her  shadow  fall  ? 

Why  in  total  Eclipses  of  the  Moon  is  she  not  invisi- 
ble, if  she  be  a  dark  opaque  body  ? 


158          Interrogations  for  Section  Twelfth.         Sec.  12 

Is  it  possible  for  the   Moon  to  be  visibly  eclipsed 
while  the  Sun  is  in  sight? 

When  the  Moon  is  full,  within  six  degrees  of  cither 
node,  will  she  be  totally  eclipsed  ? 

When  she  passes  by  the  node  in  the  earth's  shadow, 
how  much  of  the  Moon  will  be  eclipsed  1 

What  is  the  longest  time  that  the  Moon  can  suffer  an 
Eclipse  ? 

What  the  shortest  if  she  be  at  her  least  distance  1 
Why  is  this  difference  1 

What  is  the  time  of  the  longest  duration  of  an  Eclipse 
of  the  Sun  ? 

What  the  shortest,  if  the  Eclipse  be  central  ? 


SECTION  THIRTEENTH. 

SHOWING  THE  PRINCIPLES   ON    WHICH   THE    FOLLOWING 

ASTRONOMICAL  TABLES  ARE  CONSTRUCTED, 

AND  THE  METHOD  OF  CALCULATING 

THE  TIMES  OF  NEW  &  FULL 

MOONS  &  ECLIPSES 

BY  THEM. 

THE  nearer  that  any  object  is  to  the  eye  of  an  obser- 
ver, the  greater  is  the  angle  under  which  it  appears. — 
The  farther  from  the  eye,  the  less  it  appears. 

The  diameters  of  the  Sun  and  Moon  subtend  differ- 
ent angles  at  different  times.  And  at  equal  intervals  of 
time,  these  angles  are  once  at  the  greatest,  and  once  at 
the  least,  in  somewhat  more  than  a  complete  revolution 
of  the  luminary  through  the  ecliptic  from  any  given 
fixed  star,  to  the  same  star  again.  This  proves  that  the 
Sun  and  Moon  are  constantly  changing  their  distances 
from  the  earth  and  that  they  are  once  at  their  greatest 
distance,  and  once  at  their  least,  in  a  little  more  than  a 
complete  revolution. 

The  gradual  differences  of  these  angles  are  not  what 
they  would  be, if  the  luminaries  moved  in  circular  orbits, 
the  earth  beiug  supposed  to  be  placed  at  some  distance 
from  the  centre. 

But  they  agree  perfectly  with  elliptical  orbits,  suppo- 
sing the  lunar  focus  of  each  orbit  to  be  at  the  centre  of 
the  earth. 


160  On  the  Construction  of  the  follmving  Tables.  Sec.  13 

The  farthest  point  of  each  orbit  from  the  earth's  cen- 
tre, is  called  the  apogee ;  &  the  nearest  point  the  perigee. 
These  points  are  directly  opposite  each  other. 

Astronomers  divide  each  orbit  into  12  equal  parts, 
called  signs;  and  each  sign  into  30  equal  parts  called 
degrees  ,  each  degree  into  sixty  equal  parts,  called  min- 
utes,and  each  minute  into  60  equal  parts,  called  seconds. 
The  distance,  therefore,  of  the  Sun  or  Moon  from  any 
point  of  its  orbit,  is  reckoned  in  Signs,  Degrees,  Minutes 
and  Seconds.  The  distance  here  meant,  is  that  through 
which  the  luminary  has  moved  from  any  given  point, (not 
the  space  it  falls  short  thereof,)  in  coming  round  again,be 
it  ever  so  little. 

The  distance  of  the  Sun  or  Moon  from  its  apogee  at 
any  given  time,  is  called  its  mean  anomaly,  so  that  in  the 
apogee,  the  anomaly  is  nothing,  in  the  perigee,  it  is  six 
signs. 

The  motions  of  the  Sun  and  Moon  are  observed  to  be 
continually  accelerated  from  the  apogee  to  the  perigee; 
and  as  gradually  retarded  from  the  perigee  to  the  apo- 
gee ,  being  slowest  of  all  when  the  mean  anomaly  is 
nothing,  and  swiftest  when  it  is  six  signs. 

When  the  luminary  is  in  its  apogee  or  perigee,  its 
place  is  the  same  as  it  would  be  if  its  motions  were 
equable  in  all  parts  of  its  orbit.  The  supposed  equable 
motions  are  called  mean,  the  unequable  are  justly  called 
the  true. 

The  mean  place  of  the  Sun  or  Moon  is  always  for- 
warder than  the  true,  whilst  the  luminary  is  moving  from 
its  apogee  to  its  perigee;  and  the  true  place  is  always 


Sec.  13  On  the  Construction  of  the  follotmng  Tables.  161 

forwarder  than  the  mean,  whilst  the  luminary  is  mo- 
ving from  its  perigee  to  its  apogee.  In  the  former  case, 
the  anomaly  is  always  less  than  six  signs,  in  the  latter 
more. 

It  has  been  discovered  by  a  long  series  of  observa- 
tions, that  the  Sun  goes  through  the  ecliptic,  from  the 
vernal  equinox  to  the  same  again,  in  365  days,  5  hours, 
43m,  and  54s.  And  from  the  first  star  of  Ariep,  to 
the  same  star  again,  in  365  days,  6  hours,  9  minutes, 
and  24  seconds.  And  from  his  apogee  to  the  same 
again  in  365  days,  6  hours,  and  14  minutes.  The  first 
of  these,  is  called  the  Solar  year  ;  the  second  the  syde- 
real,  and  the  third  the  anamolistic  year.  The  solar 
year  is  20  minutes  and  29  seconds  shorter  than  the 
sydereal;  and  the  sydereal  year  is  4  minutes  and  36 
seconds  shorter  than  the  anamolistic.  Hence  it  ap- 
pears, that  the  equinoxial  point,  or  intersection  of  the 
ecliptic  and  equator  at  the  beginning  of  Aries,  goes 
backward,  with  respect  to  the  fixed  stars,  and  that  the 
Sun's  apogee  goes  forward. 

The  yearly  motion  of  the  earth's  or  Sun's  apogee,  is 
found  to  be  one  minute  and  six  seconds,  which  being 
subtracted  from  the  Sun's  yearly  motion,  in  longitude, 
the  remainder  is  the  Sun's  mean  anomaly. 

It  is  also  observed,  that  the  Moon  goes  through  her 
orbit  from  any  given  fixed  star  to  the  same  again,  in 
"21  days,  7  hours,  43  minutes,  and  4  seconds,  at  a  mean 
rate  ;  frcm  her  apcgee  to  her  apogee  again  in  27  days, 
13  hturs,  18  miiuites,  ar.d  <3  seccrds:  si  d  fnni  tie 
Suo  to  the  Sun  again  in  £9  days,  12  hours,  44  uai;utes. 


162   On  the  Constniction  of  the,  following  Tables.  Sec.  13 

and  3  and  ^  seconds.  This  confirms  the  idea  that  the 
Moon's  apogee  moves  forward  in  the  ecliptic,  and  that 
at  a  much  greater  rate  than  the  Sun's  apogee  ;  since 
the  Moon  is  5  hours,  55  minutes,  and  39  seconds  long- 
er in  revolving  from  her  apogee  to  her  apogee  again, 
than  from  any  star  to  the  same  again. 

The  Moon's  orbit  crosses  the  ecliptic  in  two  oppo- 
site points,  which  are  called  her  nodes,  and  it  is  obser- 
ved that  she  revolves  sooner  from  any  node  to  the  same 
node  again,  than  from  any  star  to  the  same  star  again, 
by  2  hours,  38  minutes  and  27  seconds ;  which  shows 
that  her  nodes  move  backward,  or  contrary  to  the  or- 
der of  signs  in  the  ecliptic. 

To  find  the  Moon's  mean  motion  in  a.  common  year 
of  3  Go  days,  the  proportion  is 

D      H     M      s 

As  the  Moon's  period,  27     7     43     5 

Is  to  her  whole  orbit,  or  360  degrees, 

So  is  a  common  year  of  365  days, 

To  13  revolutions  and  4s. — 9d. — 23  minutes,  5 
seconds.  The  thirteen  revolutions  are  rejected, 
and  the  remainder  is  taken  for  the  Moon's  motion 
in  365  days. 

To  calculate  the  Moon's  mean  anomaly  : 

The  Moon's  apogee  moves  once  round  her  whole 
orbit  in  8  years,  309  days,  8  hours,  and  20  minutes,  or, 
(adding  two  days  for  leap  years,)  in  3231  days,  eight 
hours  and  20  minutes.  Then, 


Sec.  13  On  the  Construction  of  the  following  Tables.  163 

As  323  Id.— 8h.— 20 

Is  to  the  whole  circle,  or  360  degrees, 

So  is  a  common  year  of  365  days, 

To  the  motion  of  the  Moon's  apogee  in  one  year= 
40  degrees,  39  minutes,  and  50  seconds. 
From  the  Moon's  mean  motion  in  longitude,  during 
one  year,  s     D    M      s 

4     9     23     5     Subtract 

the  motion  of  her  apogee,    1   10     39  50  for  the  same 
time,  and  there  remains,       2-28 — 43-15  the  Moon's 
mean  anomaly  in  one  year. 

To  find  the  mean  motion  of  the  Moon's  node  : 
The  Moon's  node  moves  backward  round  her  whole 
orbit  in  18  years,  224  days,  5  hours,  therefore  for  its 
motion  in  365  days, 

As  18  years,  224  days,  5  hours 

Is  to  the  whole  circle  or  360  degrees, 
So  is  the  year  of  365  days 

To  the  motion  of  the  Moon's  node  in  365  days=19 
degrees,  19  minutes  and  43  seconds. 
To  find  the  mean  motion  of  the  Moon  from  the  Sun. 
The  Moon's  mean  motion  in  a  common  year  of  365 
days,  is  4  signs,  9  degrees,  23  minutes  and  5  seconds 
over  and  above  13  revolutions,  and  the  Sun's  apparent 
mean  motion  in  the  same  time  is  1 1  signs,  29  degrees, 
45  minutes  and  40  seconds.     Then  from  the  Moon's 
mean  motion  for  one  year,  subtract  the  mean  motion 
of  the  Sun  for  the  same  time,  and  the  remainder  will 
be  the  mean  motion  of  the  Moon  from  the  Sun  in  one  . 
year=4  signs,  9  degrees,  37  minutes  and  25  seconds. 


164  On  the  Construction  of  the  following  Tables.  Sec.  13 

The  time,  in  which  the  Moon  revolves  from  the  Sun 
to  the  Sun  again,  (or  from  change  to  change,)  is  called 
a  lunation,  which  would  always  consist  of  29  days,  12 
hours,  44  minutes,  3  seconds,  2  thirds  and  53  fourths, 
if  the  motions  of  the  Sun  and  Moon  were  always  equa- 
hle.  Hence  12  mean  lunations  contain  354  days,  8 
hours,4S  minutes,36  seconds,  35  thirds,  and  40  fourths  ; 
which  is  10  days,  21  hours,  11  minutes,  23  seconds,  24 
thirds,  and  20  fourths  less  than  the  length  ofaccmmon 
Julian  year,  consisting  of  365  days  and  6  hours  ;  and 
13  mean  lunations  contains  383  days,  21  hours,  32 
minutes,  39  seconds,  38  thirds,  and  38  fourths,  which 
exceeds  the  length  of  a  common  Julian  year  by  18 
days,  15  hours,  32  minutes,  39  seconds,  38  thirds,  and 
38  fourths. 

The  mean  time  of  new  Moon  being  found  for  any 
given  year  and  month,  as,  suppose  for  March,  1 700, 
Old  Style ;  if  this  new  Moon  happens  later  than  the 
llth  of  March,  then  12  mean  lunations  added  to  the 
time  of  this  mean  new  Moon,  will  give  the  time  of  the 
mean  new  Moon  in  March,  1701,  after  having  thrown 
off  365  days.  But,  when  the  mean  new7  Moon  hap- 
pens before  the  llth  of  March,  we  must  add  13  mean 
lunations,  to  have  the  mean  time  of  mean  new  Mcon 
.in  March,  following,  always  taking  care  to  ?ubiracto65 
days  in  common  years,  and  in  leap  years,  366,  from  the 
sum  of  this  addition. 

Thus  in  the  year  1700,  Old  Style,  the  time  of  mean 
new  Moon  in  March,  was  the  8th  day,  at  16  hours,  1 1 
minutes,  and  25  seconds  past  four,  in  the  morning;  of 


Sic.  13  O.i  thz  Construction  of  the  following  Tables.  105 

the  9th  (by,  (according  to  common  reckoning.)  To 
this  we  must  adJ  13  mean  lunations,  from  which  sub- 
tract S  65  days,  because  the  year  1701  is  a  common 
year,  and  there  will  remain  27  days,  13  hours,  44  min- 
utes, 4  seconds,  38  thirds  and  33  fourths,  for  the  time 
of  mean  new  Moon  in  March,  in  the  year  1701. 

By  carrying  on  this  addition  and  subtraction,  until 
the  year  1703,  we  iind  the  time  of  new  Moon  in  March 
that  year,  to  be  on  tti3  6th  day,  at  7  hours,  21  minutes, 
17  seconds,  49  thirds,  and  46  fourths,  past  noon  ;  to 
which  add  13  mean  lunaiions,  and  subtract  £66  days, 
(the  year  1704  being  leap  year,)  and  there  will  re- 
main 24  days,  4  hours,  53  minutes,  57  seconds,  28 
thirds  and  20  fourths,  for  the  time  of  mean  new  Moon 
in  March,  1704.  In  this  manner,  was  the  first  of  the 
following  Tables  constructed  to  seconds,  thirds,  and 
fourths,  and  then  written  to  the  nearest  seconds. 

The  reason  why  we  chose  to  begin  the  year  with 
March,  was  to  avoid  the  inconvenience  of  adding  a 
day  to  the  tabular  time  in  leap  years,  after  the  month 
of  Feb'y.  or  subtracting  a  day  therefrom,  in  January, or 
February  in  those  years;  to  which  all  tables  of  this 
kin  1  are  sul  j-'ct,  (which  begin  the  year  with  Jan- 
uary,) in  calculating  th?  times  of  new  or  full  Moon. 

Tho  mean  an;mu!ie,-<  of  the  Sun  and  Mpon,  and  the 
Sun's  mean  motion  from  the  ascending  node  of  the 
Moon's  orbit,  are  set  down  in  Table  3d.  from  one  to 
13  lunations. 

The-e  numbers  for  13  lunations  being  added  to  the 
radical  anomalies  of  the  Sun  and  Moon ;  and  to  the 


166  On  the  Construction  of  the  following  Tables.  Sec.  13 

Sun's  mean  distance  from  the  ascending  node,  at  the 
time  of  mean  new  Moon  in  March,  1700,  (Table  first,) 
will  give  their  mean  anomalies,  and  the  Sun's  mean 
distance  from  the  node,  at  the  time  of  mean  new  Moon 
in  March,  1701  ;  and  twelve  mean  lunations  more 
with  their  mean  anomalies,  §*c.  added,  will  give  them 
for  the  time  of  mean  new  Moon,  in  March,  1 702. 

And  thus  proceed  to  continue  the  Table  as  far  as 
you  please,  always  throwing  off  12  signs,  when  their 
sum  exceeds  that  number,  and  setting  down  the  re- 
mainder as  the  proper  quantity. 

If  the  numbers  belonging  to  1700,  (in  Table  first,) 
be  subtracted  from  those  of  1800,  we  shall  have  their 
whole  differences  in  100  complete  Julian  years ;  which 
accordingly  we  find  to  be  4  days,  8  hours,  10  minutes, 
52  seconds,  15  thirds,  and  40  fourths,  with  respect  to 
the  time  of  new  Moon.  These  being  added  together 
60  times,  (taking  care  to  throw  off  a  ^0hole  lunation, 
when  the  days  exceed  twenty -nine  and  a  half,)  make 
up  60  centuries,  or  6,000  years,  as  in  Table  6th,which 
was  was  carried  on  to  seconds,  thirds  and  fourths,  and 
then  written  to  the  nearest  seconds.  In  the  same  man- 
ner were  the  respective  anomalies,  and  the  Sun's  dis- 
tance from  the  node  found  for  these  centurial  years, 
and  then  (for  want  of  room,)  written  to  the  nearest 
minutes,  which  is  sufficiently  exact  for  whole  cen- 
turies. 

By  means  of  these  two  Tables,  we  may  readily  find 
the  time  of  any  new  Moon  in  Marchy  together  with  the 
anomalies  of  the  Sun  and  Moon,  and  the  Sun's  mean 


Sec.  13  On  the  Construction  of  the  following  Tables.  167 

distance  from  the  node  at  these  times,  within  the  limits 
of  6,000  years,  either  before  or  after  the  18th  century ; 
and  the  mean  time  of  any  new,  or  full  Moon,  in  any 
given  month  after  March,  by  means  of  the  third  and 
fourth  Tables,  within  the  same  limits,  as  will  be  shown 
in  the  precepts  for  calculation. 

This  Table  is  calculated  in  conformity  to  the 
Old  Style,  for  the  purpose  of  calculating  Eclipses, 
which  have  made  their  appearances  in  former  ages, 
and  likewise  for  those,  which  will  take  place  after  the 
year  1900,  which  however,  are  easily  made  to  conform 
to  the  New  Style. 

It  would  be  a  very  easy  matter  to  calculate  the  time 
of  new  or  full  Moon,  if  the  Sun  and  Moon  moved  equa- 
bly in  all  parts  of  their  orbits.  But,  we  have  already 
shown  that  their  places  are  never  the  same,  as  they 
would  be  by  equable  motions,  except  when  they  are  in 
apogee,  or  perigee,  which  is  when  their  mean  anoma- 
lies are  either  nothing  or  6  signs.  And  that  their 
mean  places  are  always  forwarder  than  their  true, 
whilst  the  anomaly  is  less  than  6  signs  ;  and  their  true 
places  more  forward  than  their  mean,  when  the  anom- 
aly is  more. 

Hence  it  is  evident,  that  whilst  the  Sun's  anomaly  is 
less  than  9  signs,  the  moon  will  overtake  him,or  be  op- 
posite to  him  sooner,  than  she  would,if  his  motion  were 
equable  ;  and  later  whilst  his  anomaly  is  more  than  6 
signs.  The  greatest  difference  that  can  possibly  hap- 
pen between  the  mean,  and  true  time  of  new,  or  full 
Moon,  on  account  of  the  Sun's  motion,  is  three  hours, 


163  On  the  Construction  of  the  following  Tables.  See.  13 

43  minutes  and  28  seconds ;  and  that  is  when  the  Sun's 
anomaly  is  three  signs  one  degree,  or  eight  signs  and 
23  degrees;  sooner  in  the  first  ease,  and  later  in  the 
last.  la  all  other  signs  and  degrees  of  aaoimly,  the 
difference  is  gradually  less,  and  vanishes  when  the 
anomoly  is  nothing,  or  six  signs. 

The  Sun  is  in  his  apogee  on  the  30th  of  Jane,  and  in 
perigee  on  the  30th  of  December,  in  the  present  age. 
He  is  therefore,  nearer  the  earth  in  our  Winter  than 
in  Summer.  The  proportional  difference  of  distance 
deduced  frc  m  the  Sun's  apparent  diameter,  at  these 
times,  is  as  983  to  1017. 

The  Moon's  orbit  is  dilated  in  Winter,  and  ccntrac- 
terl  in  Summer,  therefore  the  lunations  are  longer  in 
Winter  than  in  Summer.  The  greatest  difference  is 
found  to  be  22  minutes  and  29  seconds:  the  lunations 
are  gradually  increasing  in  length,  whilst  the  Sun  is 
moving  from  his  apogee  to  his  perigee, anJ  decreasing 
in  length  while  he  is  moving  from  his  perigee  to  his 
apogee.  On  this  account,  the  Moon  will  be  later  ev- 
ery time  in  coming  to  her  conjunction  with  the  Sun,or 
being  in  opposition  to  him,  from  December  until  June; 
and  sooner  from  June  until  December,  than  if  her  or- 
bit ha  1  continued  of  the  same  size  dur.'n  ;  all  the  year. 

These  differences  depend  wholly  on  the  Sun's  anom- 
aly, they  are  therefore  put  together  into  one  Table,  & 
called  the  annual,  or  first  equation  of  the  mean  to  the 
true  syzygy.*  [See  Table  Seventh.]  Tiiis  equational 


*  The  word  syzygy  signifies  both  the  conjunction  and  opposition  of  the 
Sua  and  Moon. 


Sfec.  13  On  the  Constntctionofthtfottotoing  Tables.  169 

difference  is  to  b^  subtracted  from  the  time  of  the 
mean  syzygy,  when  the  Sun's  anomaly  is  less  than  six 
signs,  and  added  when  it  is  more.  At  the  greatest,  it 
is  4  hours,  10  minutes,  and  57  seconds;  viz:  3  hours, 
48  minutes  and  28  seconds,  on  the  Sun's  unequal  mo- 
tion ;  and  22  minutes  and  29  seconds  on  the  account  of 
the  dilation  of  theJMoon's  orbit 

This  compound  equation  would  be  sufficient,  for  re- 
ducing the  mean  time  of  new,  or  full  Moon  to  the  true, 
if  the  Moon's  orbit  were  of  a  circular  form,  and  her 
motion  quite  equable  in  it.  But  the  Moon's  orbit  is 
more  elliptical  than  the  Sun's  and  her  motion  in  it  so 
much  the  more  unequal.  The  difference  is  so  great, 
that  she  is  sometimes  in  conjunction  with  the  Sun,  or 
in  opposition  to  him  sooner,  by  9  hours,  47  minutes 
and  54  seconds,  than  she  would  be,  if  her  motion  wrere 
equable  ;  and  at  other  times  as  much  later.  The  for- 
mer happens  when  her  mean  anomaly  is  9  signs,  and  4 
degrees ;  and  the  latter,  when  it  is  2  signs  and  26  de- 
grees. [See  Table  9th.]  At  different  distances  of  the 
Sun  from  the  Moon's  apogee,  the  figure  of  the  Moon's 
orbit  becomes  different.  It  is  longest,  or  most  eccen- 
tric, when  the  Sun  is  in  the  sign  and  degree,  either  with 
the  Moon's  apogee,  or  perigee.  Shortest,  or  least  ec- 
centric, when  the  Sun's  distance  from  the  Moon's  apo- 
gee is  either  three  signs,  or  nine  signs;  and  at  a  mean 
state  when  the  distance  is  either  one  sign  and  fifteen 
degrees;  four  signs  and  fifteen  degrees;  seven  signs 
and  fifteen  degrees ;  or  ten  signs  and  fifteen  degrees. 
When  the  Moon's  orbit  is  at  its  greatest  eccentricity, 


170  On  the  Construction  of  the  f allotting  Tables.  Sec.  13 

Ii2r  apogeal  distance  from  the  earth's^ centre,  is  to  her 
perigeal  distance  therefrom,  as  1067  is  to  933  :  when 
least  eccentric,  as  1043  is  to  957  ;  and  when  at  the 
mean  state  as  1055  is  to  945. 

But  the  Sun's  distance  from  the  Moon's  apogee  is 
equal  to  the  quantity  of  the -Moon's  mean. anomaly,  at 
the  time  of  new  Moon  ;  and  by.  the  addition  of  6  signs, 
it  becomes  equal  in  quantity  to  the  Moon's  mean  anom- 
aly, at  the  time  of  full  Moon.  A  Table  therefore  will 
be  constructed  to  answer  all  the  various  inequalities, 
depending  on  the  different  eccentricities  of  the  Moon's 
orbit  in  the  syzygies,  and  called  the  second  equation 
of  the  mean,  to  the  true  syzygy.  [See  Table  Ninth.] 
The  Moon's  anomaly  when  equated  by  Table  Eighth, 
becomes  the  proper  argument  for  taking  out  the  sec- 
ond equation  of  time,  which  must  be  added  to  the  for- 
mer equated  time,  when  the  Moon's  anomaly  is  less 
than  six  signs,  and  subtracted  when  the  anomaly  is 
more. 

There  are  several  other  inequalities  in  the  Moon's 
motion,  which  sometimes  bring  on  the  true  syzygy  a 
little  sooner,  and  at  other  times  keep  it  back  a  little  la- 
ter, than  it  vvo^ld  otherwise  be ;  but  they  are  so  small 
that  they  may  be  all  omitted  except  two;  the  former, 
of  which  [see  Table  10th.]  depends  on  the  difference  be- 
tween the  anomalies  of  the  Sun  and  Moon  in  the 
syzygies ;  and  the  latter  [see  Table  1 1th.]  depends  on 
the  Sun's  distance  from  the  Moon's  nodes  at  these 
times.  The  greatest  difference  arising  from  the  for- 
mer is  four  minutes  and  58  seconds ;  and  from  the  lat- 


Sec.  13  On  the  Construction  of  the  following  Tables.  171 

ter  one  minute  and  34  seconds.  Having  described  the 
phenomena  arising  from  the  inequalities  of  the  solar 
and  lunar  motions,  we  shall  now  explain  the  reasons 
of  these  inequalities. 

In  all  calculations  and  observations  relating  to  the 
Sun  and  Moon,  we  have  considered  the  Sun  as  a  mo- 
ving body,  and  the  earth  as  being  at  rest  ;  since  all  the 
appearances  are*  the  same,  whether  it  be  the  Sun,  or 
earth  that  moves.  But  the  truth  is  that  the  Sua  is  at , 
rest,  and  the  earth  actually  moves  around  him,  once 
in  every  year,  in  the  plane  of  the  ecliptic.  Therefore, 
whatever  sign  and  degree  of  the  ecliptic  the  earth  is  in 
at  any  given  time,  the  Sun  will  then  appear  to  be  in  the 
opposite  sign  and  degree. 

The  nearer  any  body  is  to  the  Sun,  the  more  it  is 
attracted  by  him,  and  this  attraction  increases,  as  the 
square  of  their  distances  diminishes,  and  vice  versa. 

The  earth's  annual  orbit  is  elliptical,  and  the  Sun  is 
placed  in  one  of  its  foci.  The  remotest  point  of  the 
earth's  orbit  is  called  the  earth's  aphelion,  and  thte 
nearest  point  of  the  earth's  orbit  to  the  Sun,  is  called 
the  earth's  perihelion.  When  the  earth  is  in  its  aphe- 
lion, the  Sun  appears  to  be  in  its  apogee  ;  and  when 
the  earth  is  in  its  perihelion,  the  Sun  appears  to  be  in 
its  perigee. 

As  the  earth  moves  from  its  aphelion  to  its  perihe- 
lion, it  is  constantly  more  and  more  attracted  by  the 
Sun  ;  and  this  attraction  by  conspiring  in  some  degree 
with  the  motion  of  the  earth,  must  necessarily  acceler- 
ate its  motion. 


172  On  the  Construction  of  the  following  Tables.  Sec.  13 

But,  as  the  earth  moves  from  its  perihelion  to  its  ap- 
helion, it  is  continually  less  and  less  attracted  by  the 
Sun ;  and  as  their  attraction  acts  then  just  as  much 
against  the  earth's  motion,  as  it  has  acted  for  it  in  the 
other  half  of  the  orbit;  it  retards  iLe  n.otion  in  the 
like  degree. 

The  faster  the  earth  moves,  the  faster  will  the 
Sun  appear  to  move  ;  the  slower  the^arth  moves,  the 
slower  is  the  Sun's  apparent  motion. 

The  Moon's  orbit  is  also  elliptical,  and  the  earth 
keeps  constantly  in  one  of  its  focuses.  The  earth's  at- 
traction has  the  same  kind  of  influence  on  the  Moon's 
motion,  that  the  Sun's  attraction  has  on  the  motion  of 
the  earth.  Therefore,  the  Moon's  motion  must  be 
continually  accelerated,  whilst  she  is  passing  from  her 
apogee  to,  her  perigee  ;  and  as  gradually  retarded  in 
moving  from  her  perigee  to  her  apogee.  At  the  time 
of  new  Moon,  she  is  nearer  to  the  Sun  than  the  earth 
is  at  that  time,  by  the  whole  semi  diameter  of  the 
Moon's  orbit ;  which,  at  a  mean  state,-is  240,000  miles; 
and  at  the  full  she  is  as  many  miles  farther  from  the 
Sun,  than  the  earth  then  is.  Consequently,  the  Sun 
attaacts  the  Moon  more  than  it  attracts  the  earth,  in 
the  former  case,  and  less  in  the  latter.  The  difference 
is  greatest,  when  the  earth  is  nearest  the  Sun ;  and 
least  when  it  is  farthest  from  him.  The  obvious  result 
of  this  is,  that,  as  the  earth  is  nearest  to  the  Sun  in 
Winter,  and  farthest  from  him  in  Summer  ;  the  Moon's 
orbit  must  be  dilated  in  Winter,  and  contracted  in 
Summer,  The^e  are  the  principal  causes  of  the  differ- 


Sec.  13  On  the  Construction  of  iht  following  Tables.  173 

ence  of  time,  that  generally  happens  between  the  mean 
and  true  times  of  conjunction  or  opposition,  of  the  Sun 
and  Moon. 

The  other  two  differences,  which  depend  on  the 
difference  between  the  anomalies  of  the  Sun  and  Moon; 
and  upon  the  Sun's  distance  from  the  lunar  nodes  in 
the  syzygies,  are  occasioned  by  the  different  degrees 
of  attraction  of  the  Sun  anil  earth  upon  the  Moon,  at 
greater  or  less  distances,  according  to  their  respect- 
ive anomalies,  and  to  the  position  of  the  Moon's  nodes, 
with  respect  to  the  same. 

If  it  should  ever  happen,  that  the  anomalies  of  both 
the  feun  and  Moon,  were  either  nothing,  or  six  signs 
at  the  mean  time  of  new  or  full  Moon  ;  and  the  bun 
should  then  be  in  conjunction  with  either  of  the  Moon's 
•  nodes,  all  the  above  mentioned  equations  would  then 
vanish ;  and  the  mean,  and  true  time  of  the  syzygy, 
would  coincide  ;  but  if  ever  this  circumstance  did  hap- 
pen, we  cannot  expect  the  like  again  in  many  ages  af- 
terwards. Every  49th  lunation  returns  very  nearly  to 
the  same  tima  of  the  day  as  before ;  for  49  mean  luna- 
tions, wants  only  one  minute,  30  seconds,  34  thirds  of 
being  equal  to  1477  days.  In  2,953,059,085,108  days, 
there  are  100,000,000,000  lunations,  exactly,  and  this 
is  the  smallest  number  of  natural  days,  in  which  any 
exact  number  of  mean  lunations  are  completed. 

The  following  Tables  are  calculated  for  the  meri- 
dian of  WASHINGTON,  excepting  Table  first,  which  is 
calculated  for  the  meridian  of  LONDON,  but  they  equal- 
ly serve  for  any  other  place  by  adding  4  minutes 


174  On  ths  Construction  of  the  following  Tables.  See.  13 

to  the  tabular  time,  for  every  degree  that  the  given 
place  is  eastward  from  WASHINGTON  ;  or  subtracting  4 
minutes  for  every  degree  that  the  given  place  is  west 
ward  from  WASHINGTON. 

These  Tables  also  begin  the  day  at  noon,  and  reckon 
forward  to  the  noon  following,  for  one  day.  Thus, 
March  31st.  at  22  hours,  30  minutes,  and  25  seconds 
of  tabular  time,  (in  common  reckoning,)  will  be  April 
1st.  at  30  minutes,  25  seconds  after  ten  o'clock  in  the 
morning. 


Interrogations  for  Section  Thirteenth. 

»  . . 

Does  an  object  appear  at  a  less  angle  when  far  off, 
than  when  near  1 

Do  the  Sun  and  Moon  subtend  different  angles  at 
different  times  1 

Are  the  angles  subtended  by  the  Sun  and  Moon  once 
at  the  greatest,  and  once  at  the  least  in  one  revolution  ? 
Are  these  gradual  differences  the  same  as  they  would 
be,  if  those  luminaries  moved  in  circular  orbits  ? 

Do  they  agree  perfectly  with  elliptical  orbits  ? 

Where  must  the  lower  focus  of  each  orbit  be  placed 
to  have  them  agree  ? 

What  is  meant  by  the  term  apogee  1 

What  by  perigee  ? 


Sec.  13        Interrogations  for  Section- Thirteenth.        175 

Into  how  many  parts  do  Astronomers  divide  each 
orbit  ? 

What  is  meant  by  the  distance  of  the  Sun  or  Moon 
from  any  point  of  its  orbit  ? 

What  is  the  distance  at  any  given  point,  of  the  Sun 
or  Moon  from  its  apogee  called  ? 

What  is  the  anomaly  of  the  Sun  or  Moon  when  in 
apogee  1 

What  in  perigee  ? 

In  what  part  of  their  orbits,  are  the  Sun  and  Moon 
continually  accelerated  ? 

In  what  part  retarded  ? 
1  % 

What  are  the  mean  motions  of  the  Sun  and  Moon 
called  1 

What  are  the  unequable  called  ? 
In  what  parts  of  their  orbits  are  the  mean  motions 
forward  of  the  true  1 

In  what  part  are  the  true  forward  of  the  mean  ? 
How  many  signs  is  the  anomaly  in  the  former  case? 
How  many  in  the  latter  ? 

Does  the  Moon's  apogee  move  forward  in  the 
ecliptic  1  Does  it  move  faster  or  slower  than  the  Sun's? 

Does  the  Moon  revolve  sooner  from  any  node  to  the 
same  again,  than  from  any  fixed  star  to  the  same  again? 

If  so,  what  is  the  difference  ? 

What  is  meant  by  a  lunation  ? 

Why  Do  Astronomers  begin  the  year  with  March  ? 

What  does  Table  third  contain  ?  "what  Table  first  ? 

Why  was  Table  first  calculated  for  Old  Style  ? 


176       Interrogations  for  Section  Thirteenth.       Sec.  13 

What  is  the  greatest  difference  between  the  mean 
or  true  time  of  new  or  full  Moon,  on  account  of  the 
Sun's  motion  ? 

Are  the  lunations  longer  in  winter  than  in  summer? 

What  reason  can  you  advance  1  What  the  greatest 
difference  ?  On  what  does  these  differences  depend  '? 

What  are  they  called  ?  Why  is  this  equation  not 
sufficient  to  reduce  the  mean  time  to  the  true  ? 

Is  the  Moon's  orbit  more  elliptical  than  the  Sun's  ? 

What  is  meant  by  the  word  syzygy  1 

Is  the  Moon  sometimes  sooner  or  later  in  counjunc- 
tion  or  opposition  with  the  Sun,  than  she  would  be  if 
her  motions  were  equable  in  evefy  part  of  her  orbit  ? 

If  so,  what  is  the  greatest  difference?  On  what 
account  does  the  Mootf's  orbit  become  different  ? 

When  is  it  the  most  eccentric  ?     When  the  least  ? 

What  is  equal  to  the  Sun's  distance  from  the  Moon's 
apogee  ?  On  what  does  the  first  of  these  differences 
depend  ?  On  what  the  second  ?  V,  hat  is  the  remotest 
point  of  the  earth's  orbit  called  ?  What  is  the  nearest 
point  to  the  Sun  called?  Has  the  attraction  of  the 
earth  any  influence  on  the  motion  of  the  iVioon  1  In 
what  case  is  the  motion  continually  accelerated  ?  In 
what  case  retarded  ?  Why  is  the  Moon's  orbit  dilated 
in  winter  ?  Why  contracted  in  summer  ?  For  what 
place  are  the  following  Tables  calculated  ?  By  what 
means  do  they  serve  for  any  other  place  ? 

At  what  time  do  the  Tables  commence  the  day  ? 


SECTION  FOURTEENTH. 


Precepts  Relative  to  the  following  Tables. 

To  calculate  the  true  time  of  new  or  full  Moon,  and  Eclip- 
ses of  the  Sun  or  *\foon,  by  the  following  Tables. 
IF  the  required  new  or  full  Moon  be  between  the 
years  1800  and  1900,  takeout  the  mean  time  of  new 
Moon  in  March,  for  the  proposed  year,  from  Table  16th 
together  with  the  anomalies  of  the  Sun  and  Moon,  and 
the  Sun's  mean  distance  from  the  Moon's  ascending 
noda.  But  if  the  time  of  full  Moon  be  required  in 
March,  add  the  half  lunation  at  the  bottom  of  the  page, 
from  Table  3, with  its  anomalies,&c.  to  the  former  num- 
bers, if  the  new  Moon  falls  before  the  15th  of  March  ; 
but  if  after  the  15th  of  March,  subtract  the  half  lunation 
before  mentioned,  with  the  anomalies,  &c.  and  write 
down  the  respective  remainders. 

In  these  additions  and  subtractions,  observe  that  60 
seconds  make  a  minute,  60  minutes  make  a  degree,  30 
degrees  make  a  sign,  and  12  signs  a  circle. 

When  the  number  of  signs  exceed  12  in  addition,  re- 
ject 12,  and  set  down  the  remainder. 

IF 


178     Precepts  relating  to  Ike  following  Tables.    Sec.  14 

\Vh:  n  the  number  of  signs  to  be  subtracted  is  greater 
thin  the  number  you  subtract  from,  add  12  signs  to  the 
minued,  yon  will  then  have  a  renninder  to  set  down. 

When  the  required  new,  or  fail  Moon  is  in  any  month 
after  Mai  eh,  write  out  as  mat  y  lunations,  with  their 
anomalies,  ami  the  S  MI'S  <Vist<:iice  from  the  Moon's  as- 
eending  node  from  Table  3-1.  as  the  given  month  is  after 
M-in-h,  selling  them  regularly  below  the  numbers  ta- 
ken out  for  iYlareh  ;  add  all  these  together,  and  thev  will 
give  the  HMMIJ  llaieofihc  req  fired  new,  cr  full  Moon, 
with  the  mca.i  anornn  ies.  a. id  the  Sun's  mean  distance 
from  the  Mooii's  nsccndh  g  IK  do,  which  are  tin;  argu- 
ments for  finding  the  proper  equations 

With  thenu  nber  of  days  of  the  sum;  enter  Table  4tb 
under  the  given  month,  and  against  that  number,  you 
have  the  day  of  new,  or  full  Moon  in  the  left  hand  col- 
umn ;  which  set  before  the  hours,  minutes  and  seconds 
already  found.  But,  (as  it  will  sometimes  happen,)  if 
the  said  number  of  days  fall  sh:>rt  of  any  in  the  column, 
under  the  given  month  ;  add  from  Table  3d,  one  luna- 
tion, with  its  anomalies,  &.c.  to  tlie  aforesaid  sum,  and 
you  will  then  hav,;  a  new  s:un  of  days,  wherewith  to 
enter  Ttible  fourth,  under  the  given  month,  where  you 
are  sure  to  find  it  the  s?  c  >r.d  time,  if  the  first  fails.  With 
the  signs  and  degrees  of  the  Sun's  anomaly, enter  Table 
7th,  and  therewith  take  out  the  annual,  or  first  equation, 
for  rc:h:ci:)g  the  m  »an  to  tie  truj  s\zyg>  ;  taking  care 
t  >  make  propoit  ;>:  s  in  il.e  T  »blc  for  the  odd  minutes 
of  anomaly,  as  the  Table  gives  the  equation  only  for 
whole  degrees. 


Sec.  14    Precepts  relating  to  the  following  Tables.     179 

Observe  in  this  arid  every  other  case  of  finding  equa- 
tions, that  if  the  signs  be  \\\  the  head  of  the  Table,  their 
degrees  are  at  the  Ljft  hand,  and  arc  reckoned  down- 
wards. But  if  the  signs  be  at  the  foot  of  the  Table, 
their  degrees  arc  at  the  right  hand,  and  are  counted  up- 
wards; the  equation  being  in  the  body  of  the  Table, 
under  or  over  the  signs,  in  a  collateral  line  with  the  de- 
grees. The  terms  add,  or  subtract,  at  the  head  or  the 
foot  of  the  Tables,  where  the  signs  are  found,  show 
whether  the  equation  is  to  be  added  to  the  mean  timo 
of  new  or  full  Moon,  or  subtracted  from  it.  In  Table 
7th,  the  equation  is  to  be  subtracted,  if  the  signs  of  the 
Sun's  anomaly  be  found  at  the  head  of  the  Table;  but 
ir  is  to  be  added,  if  the  signs  be  at  the  foot. 

YViththo  signs  and  degrees  of  the  Sun's* anomaly,  at 
the  m:'un  time  of  m-\v  or  full  Moon,  enter  Table  8th, 
and  take  out  the  equation  of  the  Moon's  mean  anoma- 
ly, subtract  this  equation  from  her  mean  anomaly,  if  the 
signs  of  the  Sun's  anomaly  be  at  the  head  of  the  Table; 
but  add  it,  if  they  be  at  the  foot,  the  result  will  be  the 
M -»on's  equated  anomaly. 

With  the  signs  and  degrees  of  the  Moon's  equated 
anomaly,  enter  Table  9th,  and  take  out  the  second 
equation,  for  reducing  the  mean  to  the  trih  time  of  new 
M.)on,  adding  this  equation,  if  ihe  signs  of  the  Moon's 
equaled  anomaly  be  at  the  head  ol  the  Table  •,  but  sub- 
tracting it,  if  they  be  at  the  foot,  and  the  result  will  be 
the  mean  time  of  the  new  or  full  M  HUS,.  twice  equated. 
Subtract  the  Moon/s  equated  anomaly  from  the  Sun's 
mean  anomaly,  and  with  the  remainder,  ia  signs  and 


180     Precepts  relating  to  the  folloiving  Tables.    Sec.  14 

degrees;  enter  Table  tenth,  and  take  out  the  third  equa- 
tion, applying  it  to  the  former  equated  tim^,  as  the  titles 
add,  or  subtract  direct,  and  the  result  will  be  the  mean 
time  of  new,  or  full  Moon  thrice  equated.  With  the 
Sun's  mean  distance  from  the  ascending  node*,  enter 
Table  llth,  and  take  out  the  equation  answering  to 
that  argument ;  adding  it  to,  or  subtracting  it  from,  the 
thrice  equated  time,  as  the  titles  direct  •,  to  which  apply 
the  equation  of  natural  days,  from  Table  17th,  subtract- 
ing it,  if  the  clock  be  faster  than  the  Sun,  and  adding  it, 
if  the  Sun  be  faster  than  the  clock,  the  result  will  be  the 
true  time  of  new  or  full  Moon,  and  consequently  of  an 
Eclipse;  agreeing  with  solar  time. 

The  method  of  calculating  an  Eclipse,  for  an}'  given 
year,  will  be  shown  further  on,  and  a  few  examples  com- 
pared with  the  precepts,  will  render  the  whole  work 
plain,  and  easily  understood. 

The  Tables  begin  the  day  at  noon,  and   reckon  for- 
ward to  the  noon  following.      They  are  also  calculated 
for  the  latitude  and  longitude  of  WASHINGTON,  except- 
ing Table  first,  but  serve  for  any  place  on  the  surface 
of  the  Globe,  by  subtracting  four  minutes  for  ever}7  de- 
gree that  the  place  lies  west  of  WASHINGTON,  from  the 
true   solar  time  of  conjunction  or  opposition,  &,  adding 
four  minutes  to   the  true  solar   time  for  every   degree 
that  the  place  lies  eastward  of  WASHINGTON,  if  Table 
16th  be  used,  and  the  same  from  LONDON,  if  Table  first 
be  used,  the  result  will  be  the  true  solar  time  of  the  new 
or  full  moon,  ^consequently  of  an  eclipse  corn  sponding 
with  the  place  for  which  the  calculations  are  made* 


Sec.    14 


Astronomical  Tables. 


181 


TABLE  I.— OLD  STYLE. 

The  mean  lime,  of  new  Moon  in  March,  Old  Style;  with  the  mean  anom- 
alies of  the  Sun  and  Moon;  and  the  Sun's  mean  distance  from  the 
moon's  ascending  node,  from  the  year  1700  to  1800  inclcusive. 


Year  of 
Christ 

Mean 
moon  in 

D        H. 

new 
March 

M.      S. 

Sun's  mean 
anomaly 

S     D.       M. 

s. 

Moon's  mean 
anomaly 

S     D.       M.       S. 

Sun  smean 
tance   from 

node 
S      D.       M. 

dis- 
the 

s. 

1700 
1701 
1702 
1703 
1704 
1705 
1703 
1707 

8 
27 
16 
6 
24 
13 
2 
21 

16 
13 
22 
7 
4 
13 
22 
20 

11     25 
44      5 
32     41 
21     18 
53     57 
42     34 
31     11 
3    50 

8    19 
9      8 
8    27 
8    16 
9      5 
8    24 
8    13 
9      2 

58 
20 
36 
52 
14 
30 
46 
8 

48 
59 
51 
43 

54 
47 
89 
50 

1    22     30 
0    28      7 
11      7    55 
9    17    43 
8    23    20 
739 
5    12    57 
4    18     34 

37 

42 
47 
52 
57 
2 
7 
13 

6 

7 
8 
8 
9 
9 
10 
11 

14    31 
23     14 
1     16 
9     19 
18      2 
26      5 
4       8 
12     51 

7 
8 
55 
42 
43 
30 
17 
18 

1703 
1709 
1710 
1711 
1712 
1713 
1714 
1715 

10 
29 
18 
7 
25 
15 
4 
23 

4 
2 
11 
20 
17 
2 

11 

8 

52    27 
25      7 
13    43 
2    20 
34    59 
23     36 
12     13 
44    52 

8    21 
9      9 
8    29 
8    18 
9      6 
8    25 
8    15 
9      3 

24 
46 
2 
13 
40 
5(5 
12 
34 

43,  2    23    22 
55    2      3    59 
47    0    13     47 
3910    23     35 
51    9    29     12 
43    8      9      0 
35    6    18     48 
47    5    24    25 

18 
24 
SO 
36 

42 
47 
52 
57 

li 
0 
1 
1 
2 
3 
3 
4 

20     54 
29     37 
7     30 
15     42 

:4      2J 

10     31 

19     14 

5 

6 
54 

41 

43 

17 

13 

1716 

11  17 

33  39  8 

22 

50 

39  4   4 

14 

2  4  27 

17  05 

1717 

1   2 

22   58 

12 

6 

32  2  14 

2 

855 

19  52 

J718 

19  23 

54  45 

9 

0 

28 

44!  1  19 

39 

13 

6  14 

2  54 

1719 

9   8 

43  22 

8 

19 

44 

37  11  29 

27 

18 

6  22 

5  41 

1720 

27   6 

16  '•  1 

9 

8 

6 

4911   5 

4 

24 

8   0 

48  43 

1721 

16  15 

4  38 

8 

27 

22 

411  9  14 

52 

29 

8   8 

51  29 

1722 

5  23 

53  14 

8 

16 

38 

33  7  24 

40 

34 

8  16 

54  16 

1723 

24  21 

25  54 

9 

5 

0 

45  7   0 

17 

40 

9  25 

37  18 

1724 

13   6 

14  31 

8 

24 

16 

37 

5  10 

5 

45 

10   3 

40   5 

1725 

2  15 

3   7 

8 

13 

32 

29 

3  19 

53 

50 

10  11 

42  52 

1726 

21  12 

35  47 

9 

1 

54 

41 

2  25 

39 

56 

11  20 

25  54 

1727 

10  21 

24  23 

8 

21 

10 

34 

1   5 

19 

1 

11  28 

28  41 

1728 

28  18 

57   3 

9 

9 

52 

46 

0  10 

56 

7 

1   7 

11  42 

1729 

13  3 

45  40 

8 

23 

48 

39 

10  20 

44 

12 

1  15 

14  29 

1730 

7  12 

34  16 

8 

18 

4 

31 

9   0 

32 

17 

1  23 

17  16 

1731 

26  10 

6  56 

9 

6 

26 

42 

8   6 

9 

23 

3   2 

0  17 

1732 

14  18 

55  33 

8 

25 

42 

34 

6  15 

57 

28 

3  10 

3   4 

1733 

4   3 

44   9 

8 

14 

58 

26 

4  25 

45 

33 

3  18 

5  51 

1734 

23   1 

16  49 

9 

3 

20 

38 

4   1 

22 

39  4  26 

48  53 

1735 

12  10 

3  25 

8 

22 

36 

30 

2  11 

10 

44 

5   4 

51  40 

1736 

0  18 

54   2 

8 

11 

52 

22 

0  20 

58 

49 

5  12 

54  27 

1737 

19  16 

26  42 

9 

0 

14 

34 

11  26 

35 

55 

6  21 

37  29 

1738 

0   1 

15  18 

8 

19 

30 

26 

10   6 

21 

0 

6  29 

40  16 

182 


Astronomical  Tabks. 


Sec.  1 4 


TABLE  I.— OLD  STYLE, 

CONTINUED. 


Year  of 

Christ 

Mean  new 
moon  in  March 

I>   H.   M.   S. 

Sun's  mean 
anomaly 

S   D.   M.   S. 

Moon's  mean 
anomaly 

S   D.   M.S. 

Sun's  mean  dis- 
tance fro  ;n  tho 

node. 
1  S.   P.  M,   S. 

17d9 

1710 
1741 
1742 
1743 
1744 

21  22 
16   7 
5  16 
24  13 
13  22 
2   7 

47 
36 
25 
57 
45 
35 

58 
34 

11 
52 
127 
4 

9   7 
8  27 
8  16 
9   4 
8  24 
8  13 

52 

8 
24 
46 
2 
13 

3:. 
30 
27 

o; 

27 
20 

9  1-2 
7  21 
6.  1 
5   7 
3  17 
1  28 

1   6 
49  11 
37  16 

14  2-2 
2  27 
50  32 

8   8 
8  16 
8  24 
10   3 
10  18 
10  19 

23 
26 
18 
11 
14 
17 

18 
5 
52 

54 
44 

28 

1745 
1746 
1747 

1748 
1749 
1750 
1751 

21   5 
10  13 
29  11 

17  20 
7   5 
26   2 
15  11 

7 
56 
23 
17 
6 
33 
27 

44  9   1 
23,  8  23 
099 
36  8  28 
13  8  17 
53  9   6 
29  8  25 

40 
56 
18 
34 

53 
12 

23 

bl: 
24 
30 
2 
20 
3i 
24 

1   2 
tl  12 
U  17 
8  27 
7   7 
6  13 
4  22 

27  38 
15  4-J 
52  4!'. 
43  54 
23  50 
6   5 
54  1  ) 

11  23 
0   6 
1  14 
1  22 
,2   0 
3   9 
3  17 

0 
3 

46 
49 
51 
34 
37 

30 
17 
13 
5 
52 
5J 
40 

1752 
1753 
1754 
1755 
1756 
1757 
1758 

1759 
1760 
1761 
1762 
1763 
1764 
1765 
1766 
1767 
1763 
1769 
1770 
1771 
1772 
.1773 

3  23 
22  17 
12   2 
1  11 

19   8 
8  17 
27  15 

16 
48 
37 
25 

53 
47 
19 

6  8  14 
45  9   3 
22  8  22 
59  8  11 
38  9   0 
151  S  19 
54  9   7 

44 
6 
22 
33 
0 
16 
33 

1: 

2>- 

23 

£ 

1(5 
2S 

3   2 
2   8 
0  13 
13  27 
10   3 
8  13 
7  18 

42  LJ 
19  21 
7  26 
55  31 
32  37 
23  42 
57  43 

3  25 
5   4 
5  12 

5  20 
6  29 

7   7 
8  15 

43 
23 
26 
23 
12 
14 
57 

27 
28 
15 
2 

3 

50 
52 
i>9 
26 
27 
14 
1 
2 

49 

17   0 
5   8 
24   6 
13  15 
3   0 
20  21 
10   6 

6 

57 

29 
18 
7 
39 
28 

31 
8 

47 
24 
1 

40 
17 

d  23 
8  16 
9   4 
8  23 
8  13 
9   1 
8  23 

64 
13 
32 
48 
4 
26 
42 

20 
12 
24 
16 

8 
20 
13 

5  2j 
4   3 
3  14 
1  23 
0   3 
11   9 
9  19 

45  o4 
34   0 
11   G 
59  11 
47  16 
24  21 
12  26 

8  24 
9   2 
10  10 
13  13 
10  26 
0   5 
0  13 

0 
3 
46 
49 
52 
35 
37 

29   4 
IS  12 
6  21 
25  19 
15   3 
4  12 
22  10 
11  19 

0 

49 
38 
10 

59 
43 
23 
9 

66  9   9 

33  8  28 
10  8  17 
40  9   5 
2-3  8  25 
2  8  14 
43  9   2 
19  8  22 

4 
23 
36 
58 
14 
33 
52 
8 

23 
17 
9 
21 
13 
5 
17 
9 

8  24 
7   4 
5  14 
4  23 
2  29 
1   9 
0  15 
10  25 

4-J  32 
37  3? 
25  42 
2  48 
53  53 
33  53 
16   4 
4   9 

1  22 
2   0 
2   8 
3  17 
3  25 
4   3 
5  11 
5  23 
5  28 
7   6 
7  14 
8  23 
9   1 
9   9 
10  18 
10  2S 

20 
23 
26 
9 
12 
15 
58 
0 

51 
33 
25 
27 
14 
1 
3 
53 

1774 
1775 
1776 
1777 

1778 
1779 
1783 
1781 

1   3 

23   1 
8  10 
27   7 
16  16 
6   1 
23  23 
13  7 

o/ 
30 
1-9 
51 
40 
29 
1 
50 

63 

2-3 
12 

51 
28 
4 
44 
21 

8  11 
8  29 
8  19 
9   7 
8  26 
8  15 
8   4 
8  23 

24 
46 
2 
24 
40 
56 
18 
34 

1 

ll 

17 
9 

1 
13 
5 

y   4 
8  10 
6  20 
5  25 
4   5 
2  15 
1  21 
0  0 

02  14 
29  23 
17  25 
54  31 
42  36 
30  41 
7  47 
£5  52 

•6 
49 
49 
32 
35 
88 
21 
23 

b7 
88 
25 
26 
13 
0 
1 
4$ 

Sec.  14 


Astronomical    Tables. 


183 


TABLE    I.— OLD  STYLE, 


CONCLUDED. 


Year  of 

Christ 

Mean   new 
moon  in  March 

r>      ii.     M.    s. 

I     Sun's  mean 
anomaly 

Moon's  mean 
anomaly 

s    r».     M.     s. 

Sun's  mean 
tance  from 

node 
S.     D.      M 

dis- 
the 

.     s 

i7«J  , 
1783 
1781 
1785 
1783 
1787 
1788 

2 
21 

9 
23 

18 
7 
2-> 

id 
14 
23 
20 
5 
14 
11 

3d    J7 
11     37 
0     13 
32    53 
21     10 
10      6 
42    46 

3    12 
9      1 
8    20 
9      8 
8    28 
8    17 
9      5 

49 
12 
28 
50 
6 
21 
44 

58 
10 
3 
1,5 

7 

59 
11 

10      U 

9    16 
7    26 
7      1 
5    11 
3    21 
2    26 

43    57 
21       3 
9      8 
46     U 
34     U 
22    24 
59     30 

11 
0 
0 
1 

2 
2 
3 

4 
4 
5 
5 

4    26 
13      9 
21     12 
29    55 
7    58 
16      0 
24    44 

35 
36 
23 
25 
12 
59 
1 

i7Bj 

1790 
1791 
17P2 

14 
4 
23 

;! 

20 
5 
2 
11 

31     £3 
19    :9 
52     19 

41     15 

3    2o 
8    14 
9      2 
8    21 

0 
15 
38 
53 

3 
56 

SB 

i      6 
11    16 
10    22 
9      2 

47     c5 
35     40 
12     <Q 
0     £2 

2     46 
10    49 
19     32 
27     T5 

48 
S5 

17 
£4 

1793 

1794 
1795 
1796 

3J 
19 

9 
27 

9 

18 
2 

0 

13     55 
2     ,2 
5i       S 
2:*     '8 

9     .0 

3    29 

8     i8 
9      7 

10 

32 

47 
10 

j  . 
3 
ffi 

7 

8      7 
6     17 
4    27 
4      2 

~2      2~ 
0    22 

*    28 
O      7 

b7    £8 
26      4 
14       9 
51      14 

7 
7 
7 
9 

6     18 
14    21 
22     24 
1       7 

£6 
13 
0 
1 

~48 

35 
37 
24 

1797 
17i'8 
1799 
1800 

.0 
5 
24 
3 

y 
18 
15 
0 

12     -4 
1        1 
23     41 

22     <7 

8    2o 
8     J5 
9      4 
8    °3 

25 
41 
4 
'9 

,9 

5i 

1 

39     .< 
27     2; 
4     3 
52     31 

9 
9 
0 

±4 

9       9 
17     12 
25    55 
3    58 

184 


Astronomical  Tablet. 


See.  14 


TABLE  II. 

The  first  new  Moon,  with  the  mean  anomalies  of  the 
Sun  and  Moon,  and  the  Sun's  mean  distance  from  the 
ascending  node,  next,  after  complete  centuries  of 
Julian  years. 


Julian 
Years. 

D     H    M     S 

S  D  M 

S    D     M 

S     D    M 

100 
200 
300 
400 
500 

4       8     10     5* 
8     16     21     44 
13      0    32    37 
17       8    43    29 
21     16     54    21 

0  3  21 
0  6  42 
0  10  3 
0  13  24 
0  16  46 

8     15     22 
5       0    44 
1     16       6 
10       1     28 
6     16     50 

4     19     27 
9       8     55 
1     28     22 
6     17     49 
11       7     16 

600 
700 
SOO 
SOO 
1000 

26       1       5     14 
0    20    32      3 
5      4    42     55 
9     12    53    47 

13    21       4     40 

0  20  7 
11  24  22 
1  1  27  43 
0  1  4 
0  4  25 

3       2     12 
10     21     45 

777 
3    22    2£ 
0      7     51 

3     26     44 
7     15     31 
0       4     58 
4     24     25 
913     53 

2000 
3000 
4000 
5000 
6000 

-27     18       9     It* 
12      2    29     56 
25     23     34     35 
10       7     55     12 

24       4     53     52 

0  8  50 
11  14  8 
11  18  33 
10  23  52 
10  28  17 

0     15    42 

11     27     43 
0       5     34 
11     17     36 
1  1     25     27 

6     27     45 
3     10    58 
0     24     50 
983 
6     21     56 

Sec.  14 


Astronomical  Tables. 
TABLE  III. 


Mean  anomalies  of  the  Sun  and  Moon,  and  the  Sun's  mean  distance  from 
the  node,  for  13  and  £  mean  lunations. 


3  2 

i  9 

Is- 

mean  lunations. 

D         H        M        S 

Sun's  mean  anomaly 

S         T>        M        S 

Moon's  mean 
anomaly. 
SUMS 

Sun's  mean  dist. 
from  the  node. 

S        D        M         S 

i 

2 
fr 
4 

i 

29     12    44      3 
59       1     28      6 
S3     14     12       9 
118       2     56     12 
147     15     40     15 

0    29       6     19 
1     28     12    39 
2     27     18     58 
3     26     25     17 
4     25     31     37 

0    25    49       0 
1     21     38       1 
2     17    27       1 
3     13     16      2 
4952 

1       0    40     1- 
2       1     20    2S 
3      2      04'. 
4      2    40    56 
5      3    21     1C 

6 
7 
8 
9 
10 

177       4    24     18 
206     17       8     21 
233       5     52     24 
265     18     36     27 
295       7     20    30 

5     24     37    56 
6     23    44     15 
7     22     50     35 
8     21     56     54 
9    21       3     14 

5      4    54      3 
6      0    43      3 
6    26    32      3 
7     22    21       4 
8v    18     10      4 

6      4      1     24 

^7      4    41     3t 
8       5    21     5i 
9626 
10    .6    42    2^ 

r 

12 
13 

324    20      4    33 
354       8     48     36 
383    21     32    40 

10    20      9    33 
11     19     15    52 
0     18    22     12 

9     13    59      5 
10      9    43      5 
11       5     37       6 

11       7    22    34 
0      8       24- 
1       8    41       1 

"i"  _ 

3    C 
'      » 

14     13    22      2 

0     14    33     10 

6     12     54     30 

0    15    20       7 

(JC^  The  half  lunation  above,  is  used  in  finding  the  mean  time  of  fall 
JJ/ocm,  and  likewise  in  calculating  her  Eclipses. 


186 


Astronomical  Tables. 
TABLE  IV. 


Sec.  14 


The  days  of  the  Year,  reckoned  from  the  beginning 
of  March. 


— 

0 

— 

1 

s 

c 

3 

|| 

CO 

"S. 
2" 

O 

| 

o 

a 

C 

9 

EM 

to 

s 
c 
a 

f 

fft 

5 

*T 

E 

5 

ff 

f1 

* 

?" 

c- 
n 

~~[ 

j 

32 

62 

93 

123154 

165215 

246276 

307  M* 

2 

33 

63 

91 

186216 

247.;:77 

308:3:;9 

3 

g 

34  64 

95 

125  1?6 

187  217 

248  278 

309 

34t> 

4 

4 

35 

65 

ye 

126157 

188il>18 

2491279 

3!0 

34! 

5 

5 

J6 

66 

97 

'27  ir;8 

189  219 

250*280 

31  ! 

342 

~g 

~6 

^ 

— 
67 

~98 

128  [59 

F90 

'"^m 

251281 

••~7:2 

543 

7 

7 

i8 

99 

129 

160 

191 

221 

252  2f  2 

313 

344 

8 

8 

i9 

69 

100 

J3o 

161 

192 

222 

253  283 

3!4 

345 

9 

10 

70 

10! 

I3i 

193 

223 

254:284 

315 

3^6 

10 

H 

n 

71 

102 

132 

163 

194 

224 

255  285 

3'6 

347 

n 

1 

12 

72 

108 

13;\ 

164 

19f> 

225 

256  l;86 

317 

34? 

12  1- 

[:^ 

73 

H>4 

!34 

165 

196 

226 

257  287 

318 

349 

13  |f 

'1 

74 

105 

i  3"> 

166 

197 

227 

258  288 

3!9 

350 

14'  1 

75 

106 

136 

l<>7 

198 

228 

259 

289 

320  361 

.'5  1 

6 

76 

107 

137 

168 

199 

219 

26(. 

290 

an  aw 

Hi  i 

7 

77 

108 

538 

769 

200 

23T> 

261 

291 

322363 

1  7 

I 

8 

78 

109 

139 

170 

201 

•231 

262 

292 

323  354 

18 

! 

r9 

79 

HO 

140 

171 

202 

232 

263 

293 

324  '355 

19 

•i 

;> 

H' 

111 

141 

172!  203 

233 

264 

294 

325  356 

21) 

.' 

>i 

81 

112 

;42 

173 

204 

234 

265 

295 

326  357 

21 

'2. 

>2 

-s- 

113 

143 

174 

205 

235  26<i 

296 

327  :--58 

22 
23 

22 

2 

53 
54 

83 

84 

114 
'15 

144 

175  206 
176207 

236 
237 

267 

268 

297 

298 

328  359 
329  360 

24 

24 

55 

85 

1  16 

46 

177208 

238 

269 

299 

?3t  '  36  ! 

25 

2: 

5fi 

86 

117 

147 

178J209 

239 

270 

3oo 

331362 

§62657 

87 

118 

148 

179|2l(> 

240 

27  1  30  1  1:s32  863 

27;27<58 

S8 

119 

49 

180J211 

241 

272  802SS3364 

28  28  59 

89 

120 

150 

181  212 

242 

27313031334  365J 

29:29  60 

:)) 

121 

151 

i  82  2  i  3  248 

274304^335 

30 

3061 

9' 

122 

152 

183 

2141244 

275 

3  5;33f> 

81 

31  00 

9- 

153 

184 

(245 

306:337 

Sec.  14 


Astronomical  Tables. 
TABLE  V. 

THE  SUN'S  DECLINATION, 


187 


ARCrUMElVT.«Tlie  Sun's  trne  place. 


De- 
grees 

Signs. 
0.    N. 
6    S. 

Signs. 
1.    N. 
7     S. 

Signs. 

,2.    N. 
8     S. 

De- 
grees 

U.    M. 

U.    M. 

D.    M. 

0 
1 
2 
3 

4 
5 

0   0 
0  24 
0  48 
1   12 
1  36 
1  59 

11   30 
11   51 
12   11 
12  32 
12  53 
13   13 

20   11 

20  24 
20  36 
20  48 
20  59 
21   10 

30 
29 
28 
27 
26 
25 

6 
7 
8 
9 

10 

2  47 
3   11 
3  34 
3  58 

13  33 
13  53 
14  12 
14  31 
14  50 

21  21 
21  31 
21   41 
21   50 
21   59 

24 
23 
22 
2t 
20 

1  1 

12 

13 
14 
15 

4  22 
4  45 
5   9 
5  32 
5   55  t 

1  5   9 
15  28 
15  46 
16   4 
16  22 

•22    8 
22   16 
22  24 
22  31 
22   38 

19 

18 
17 
16 
15 

16 
17 

18 
19 
20 

6   18 
6  41 
7   4 
7  27 
7  50  k 

16  39 
16  57 
17  14 
17  30 
17  -46 

22  45 
22   51 
22  56 
23   2 
23   6 

i4 

IS 
12 
11 
10 

2  -' 
23 
2-1 
25 

8   13 
8  35 
9  57 
9  20 
9  4' 

18   2 
18  18 
18  33 
18  48 
19   3 

23   11 
23   14 
23   18 
23  21 

22  23 

9 
8 
7 
6 
5 

'  '  I  ! 

17 

28 
29 

30 

10   4 
10  25 
10  47 
11    8 
I  \   SO 

19  17 
19  31 
19  45 
19  58 

20  1  ! 

23  25 
23   27 
23  28 
23  29 
23  29 

4 
3 
2 

1 
0 

.  c- 

Signs. 

Signs. 

Si  ;n  -•. 

De- 

gtees. 

11    S. 
5    N. 

10    IS. 
4    N. 

y   s. 

S    N. 

grees. 

188 


Astronomical  Tables. 


Sec.  14 


TABLE  VI. 

Equation  of  the  Sun's  centre,  or  the  difference  between  his  mean  and 

true  place. 


ARGUMENT—  Sun's  mean  anomaly 


[f 

i  ' 

Subtract. 

P 

| 

<» 

rj 
0) 

0  Signs. 

D      M        S 

1  aJ.ga. 

D     M        S 

2^.  gas.   -  3  sSi^us. 
DM        S  |D      M        s 

4  Signs-. 

D       M         S 

o  fSi.rio. 
n    *M       s 

0 

1 

2 
3 
4 
5 

0      0    00 
0      1     59 
0      3    57 
0      5    o 
075 
095 

U     56     47 
0    58     30 
0     12 
[       1     55 
[      3    3 
[       5     11 

I     3J       6 
I    40      7 
[     41       t 

I     42       S 
[    42    5' 
(     4:{     o 

1     55     37 
55     3 
55    38 
55     36 
55     31 
55 

1       41        U 

1     40     i 
1     39     1C 

38      6 
I     37       C 

•     :  -'i     .5i. 

'.,        03        JO 

0    57.      7 
0     55     19 

0     t.J    30 
0    51     40 
0    49     40 

i»U 

29 
28 
27 
It 

25 

o 
7 

8 
9 
10 

OH. 
0     13    4t 
0     15    46 
0     17    4. 
0     19     4' 

i         .,      0\J 

I      8    ?. 
I     10       i 
I     11     3( 
[     1  ;       9 

L     44     4 
[     45     34 
f     43     2 
I     47       8 
[     47       L 

55     15 
55     0, 
54     5C 
54    33 
54     17 

i      Ji     43 
1     33    3i 
1     32     1'.; 
1     31       4 
1     29     47 

U      17      j'i 
i    46     05 
•»    44     11 
0    42     16 
0    40     21 

H4 

23 
22 
21       I 
20 

11 
12 
13 
14 
15 

0    21     jt 
0    23    33 

0    25     29 
0     27     25 
0.   29     20 

i     14     41 
I     16     11 
[     17     40 
I     19       8 
f     20    34 

I     49     15:1     53     3o 
I     49     541     53     12 
I     50    301     52     46 
T     51       fcjl      52     IS 

27       9 
25     4F 
24     2 
23      0 

0     Jj     -Ji 
)     33     28 
)    34    30 
)    32     32 
0     30     33 

tJ 

13 
17 
16 
15        ! 

16 

17 
18 
19 
20 

0     31     15 
0    33      9 

0    35      £ 
0    3S     5- 
0    38    47 

i     21     59 

I    23    2-2 
I     21    44 
I     26       5 

I     27    2^ 

I     51     a, 
I     52       8 
I     52    3^ 
[     53      : 
I    53    27 

£     Ji     43 
51      15 
I     50    41 

[     50       5 
[     4)     26 

21     31 
20       6 
IS     3 
17       5 
33 

J     -28     33 
0     26     33 
3     24     3J 

)     -11    32 
0    20    30 

U 

13 

w 

ir 
10 

21 
22 
23 

24 

25 

0    40    39 
0    42    30 
0    44    20 
0    43       S 
0    47     57 

I     28     4 
L     29     57 
[    31     1, 
[    32    2£ 
t     33     3 

I     i>J     6L 
[     54     H 
I     54     28 
L     54     4 
f     54     5 

[     *,     -. 
48       3 
[     47     IS 
[     46     3-. 
I     45     4 

U     60 

12    2  : 
10    47 
9       i, 

7     9 

>     18     2d 
0    16    28 
0    14    24 
J     12     21 
1     10     i 

8 
7 
6 
5 

-)  ."' 

27 
23 
29 
90 

0    49    451 
0     51     32  i 
0    53     18 
0     55      3 
0    56    47 

I     34    45 
I     35     53 
(     33     51 
I    38      8 
[    39       6 
10  Signs. 

I     53     1 
[     55     20 
[     55     28 
I     55     3 

[     b>     37 

9  Sterna 

L     44     .3' 

r     44       1 
r     4$       7 
[     42     lOj 

I     41     12 
8  Si>rns. 

L                  J            'i 

[       4       7 
[       2     24 
0    3. 
0    58     53 

i)       611* 
047 
024 
TOO 

4 
3 
2 

0 
J?    ' 

•^ 

• 

| 

o 
<s 

• 

11  Si-rns. 

7  Si  jns. 

6Si~ns. 

Add. 

Sec.  14  Astronomical    Tables.  183 

TABLE  VII. 

The  annual,  or  first  equation  of  the  mean,  to  the  true  syzygy. 

ARGUMSNT-»Su»'B  mean  anomaly* 


1 
2 
3 
4 
_5_ 

IF 

7 

8 

9 

U 

IT 

12 

13 

14 

15_ 

16 

17 

13 

19 

2J_ 

2T 

22 

2J 

24 

2J_ 

23 
27 
23 

29 


Subtract. 

0  Signs. 

.1       AC        S 

~J      0      C 
0       4     IS 
0       8     3-j 
0     12    51 
0     17       b 
JO     21     24 

I   1  Sign. 
|H     M      s 
2       3~12 
2       b     50 
2     U     36 
2     14     14 
2     17     52 
2     21     27 

2  Signs.      3  Signs. 

i-I       JU        S    i  J       31       S 

,3~35       04~10~53 
J     37     104     10     5V 
3    39     Id  4     10    55 
3     41     234     10     4o 
3     43     264     10     b9 
3     45     254     10     24 

4  Signs. 

H       M       S 

3~~39~~30 
3     37     19 
3     35       6 
3     32     5J 
3     3J     30 
3     23       5 

5  Signs. 

H  M  S 

2  7~~45 
2  3  55 
2  0  1 
1  56  5 
1  52  6 
1  48  4 

0     25     39 
J     2.3     55 
0     34     11 
J     33     2o 
0     42     2y 

0     46     5^ 
0     51       4 
0     5J     1; 
0    59    2? 
1       3     3o 

2     2j     OJ 
'2     28    2fa 
-i     bl     5t 
1     36     22 
2     33     44 

3    47     194     10       43     2a     36 
3    49      74       9     393     23       0 
3     50    5J  4       9     10  3     20     20 
3     52     29'4       8     b/  3     I/     3o 
3  '  54       41      7     59  3     14     4^ 

1  41  1 
1  39  56 
1  35  49 
1  31  41 

i  27  31 

2     42       b 
2     45     lo 
2    43     30 
2     51     40 
2     54     4s 

3     55     35 
3    57      2 
3     o3     2? 
J     59     49 
3       1       7 

4       2     18 
4       3     23 
4       4     2J 
4       5     U 
4       6     10 

4       7     163     11     5(J 
4       6     293       9       6 
4       5     373       6     10 
4       4     413       3     10 
4       3     403       0       7 

1  23  1G 
1  19  5 
1  14  49 
1  10  33 
1  6  15 

1       7    45 
1     11     5* 
1     16      0 
1     2J       6 
1     24     10 

~IdT~T2 
1     32     12 
i     36     K 
1     40       b 
1     4i       1 

2     57     55 
3       0     6-i 
3       3     51 
3       6     4d 
3       9     36 

4       2     3o 
4       1     2o 
4       0     1^ 

j     53     ij- 
3     57     *, 

2     57      0 
2     53     4b 
2    5D     3b 

2     47     lo 
J     43     5, 
2     4J     ij3 
J    ^7      0 
2     b3     3o 
•2     3J       2 
2     26     26 

i  1  56 
J  57  3b 
J  53  la 

J  48  62; 
J  44  lidi 
J  iO  2 
J  bo  be 
J  bl  10 
J  26  4^ 
J  22  l?| 

3     12     24 
6     15       9 
3     17     51 
3     2-J     3J 
3     23       L, 

4       6     56 
4       7     41 
4       8     21 
4       8    ov 
4       9     29 

3     5J     o^ 
3     54     2u 
3     oJ     4^ 
u     ^1       9 
3     49     26 

i     47     ^4 
i     51     46 
i     55     37 
1     5)     26 
2       3     12 

3     25     36 
3     23       b 
3     30     26 
3     32     4o 
3     35       0 

4       9     55J3     47     3s 
4     10     16|3     45     44 
4     10    333    43    45 
4     10     453     41     40 
4     10    533    30     30 

2     22     470     17   .50 
'2     19       5,0     13     2o 
2     15     20  J       8    56 

2     11     350      4    2b 

2      7    450      0      0 

11  Signs.  1  10  Signs. 

9  Signs. 

8  Signs. 

7  Signs. 

6  dgns. 

Add. 

30 

29 
28 
27 
26 
J5 

24 
23 
22 
21 
J20 
19 
18 
17 
.16 
_15 

14 
13 
12 
11 
10 

~9 
8 
7 
6 

_5 
4 
3 
2 
1 
0 

1? 


Astronomical  Tables.  Sec.    14 

190 

TABLE  VIII. 

EQUATION    OF    THE    MOON?S    MEAN    ANOMALY. 
ARGUMENT— Suu's  mean  anomaly. 


S? 

1 


0 

1 

2 
3 

4 

_5_ 

6 

7 

8 

9 

12. 

11 

12 

13 

14 

IfL 
16" 
17 
18 
19 
20_ 

21 

22 

23 

24 

25^ 

26 

27 

28 

29 

30_ 

S? 


Subtract. 


0  Signs, 


1  S  gn,     |  2  Signs,   i  3  Signs,  j  4  Signs,   j  5  Sigus. 

SjH       M         S    H      M         S   !H      M        S     H      M         SUM         S 


|0      0.0 

0     46     451     21     32 

1     35       111     23       40     48     19 

30 

0       1     37 

0    48     10,1     22    21 

1     35      2 

1     22     140    46    51 

29 

0      3     13 

0     49     34:1     23     10 

I     35       1 

i     21     240     45     23 

28 

0      4    520     50    531     23    57 

1     35      0 

1     20     320     43     54 

27 

0      6     280     52     191     24    41 

1     34    57 

1     19     380     42    24 

26 

10      8      60     53     40 

1     25    24 

1     34     50J1     18     420     40     53 

25 

0      9    42:0    55      0 

1     26      6 

1     34     43  1     17     45  0     39     21 

24 

0     11     200    56     21 

1     26     48 

1     34     33 

1     16     480     37    49 

23 

0     12    560    57     38 

1     27    28 

1     34     22 

1     15    470     36     15 

22 

0     14     330    58    56 

1     28      6 

1     34       9 

1     14     440     34     40 

21 

0     16     101       0     13 

1     33    431     33    53 

1     13     410     33      5 

20 

0     17     471       1     29 

1     29     17 

1     33     371     12     370     31     31 

19 

0     19     231       2    43 

1     29     51 

1     33    201     11     330     29     54J  18 

0     20    59 

1       3     56 

1     30    22 

1     33      0 

1     10    260     28     IS 

17 

0     22     35  1       5       8 

1     30     50 

1     32     38 

1       9     170     26     40 

16 

0     24     101       6     IS 

1     31     19 

1     32     14 

1       8      8 

0     25       3    15 

0    25     451       7     27 

I     31     45 

1     31     50 

1       6     58 

0    23     23 

14 

0    27     19  1       8     36 

i     32     12 

1     31     23 

1       5     46 

0    21     451  13 

0    28    521       9     42 

1     32     34 

1     30     55  1       4     32 

0    20      7 

12 

)     30    25 

1     10     49 

1     32    57|1     30    25  1       3     19 

0     18    28 

H 

)     31     57 

1     11     54 

1     33     17 

1     29     54 

1       2       1 

0     16     48 

10 

0     33    29 

I~12     58 

r~33~~36 

1  "29     20 

1       0     45 

0     15      8 

9 

0     35      2 

1     14       1 

1     33    52 

1     28    45 

0    59     26 

0     13    28 

8 

)     36     32 

1     15       1 

1     34      6 

1     23       9 

0    58      7 

0     11     48 

7 

,)     33       1 

1     16      0 

1     34     18 

1     27     300     56     45 

0     10      7 

6 

0     39     29 

1     16    59 

1     34    30 

1     26     5010     55     23 

0      8     20 

5 

0     40     59 

1     17    57 

1     34     40 

1     26     27 

0     54       1 

0       6     44  |     4 

0    42     26 

1     18    52 

1     34     48 

1     25       5 

0     52     37 

053 

3 

0     43     54 

1     19     47 

1     34    541     24     39 

0     51     11 

3       3    21 

2 

0     45     19 

I     20     40 

1     34    58 

1     23     52 

0     49     45  0       1     40 

1 

0    47    45 

1     21     32 

1     35       1 

1     23      4 

0     48     19 

J      0       C 

0 

11  Signs. 

10  Signs. 

9  Signs. 

8  Signs. 

7  Signs 

6  Signs. 

1 

Add. 

i 

Sec.  14  Astronomical  Tables.  291 

TABLE  IX. 

The  second  equation  of  the  mean,  to  the  true  syzygy. 

.      -  i  — • 

ARCrUMENT««Moon'8  equated  anomaly* 

~  Add 


1 

0 
Signs. 

H.  M.  S. 

1 

Si^ns. 

H.        M.         S. 

•2 

Si?ns. 

H.         M.         S 

3 

Signs, 
a.      in.      s. 

4 

Signs. 

K.        M.        S 

5 

Signs. 

H.        M.        S. 

S 

0 

2 
3 
4 
5 

0  0  (' 

)  10  58- 
»  21  5h 
0  32  54 

o    43    : 
o    r»4    51 

>      12     48 
.1     2  1      5f> 
.»     30     57 
.1     39      M 
:»     48     3? 
">      57      1  7 

8      47        « 
8      51      4f 
8      56      H 

9        0     2; 
9       43! 
9         fi      0f 

9      46      44 
9     45       3 
9     45      12 
3     44      11 
•)     42     59 
D     41      31 

3        b      59 
8        3     12 
7     57     2S 
7     51      35 
7     45     4f 
7     39     41 

t     34     33 
4     26        1 
4     17     25 
4       8     47 
407 
3     51      23 

30 

29 
28 

27 
26 

25 

6 

7 
8 
9 
10 

1  5  4: 

1  18  41: 

1  27  44 
I  38  4<; 
I  49  3/3 

•'>        5      5  1 
o      14      19 

8     22     4  1 
i      30      57 
u      39        4 

9      12        9 
9      1  5     4H 
9      19        5 
9     22      M 
i)      23      1  % 

:)      40        8 
9      38       1  9 
9      36     24 
.>      34      1! 
9     32        1 

;      33     3; 

7     27     2£ 
7     21 
7      14     3* 
7        7      5' 

3     4s     3- 

}     33     Sa 
;     24     4£ 
3      1  5     41 
3        6      4: 

24 
23 
22 
21 
20 

11 
in 

13 
14 
15' 

2  0  23 
2  11  1<> 
2  21  54 
•2  32  3  i 
2  43  9 

i      47        » 
">      54     4' 
7        2     24 
7        9      5JC 
7      17        H 

:>       27        04 

t      30      3 
9     32     5; 

)    35     i  «•: 

)      37      14 

.»     29     33 
:J     26      54 
.)     24       4 
J     21        3 
J      17      51 

7         I        i 

0      54        1; 
6      47        {• 
ij      40        t 
H      32      5f 

2      57      43 
2     48     39 
2     39     31 
2     30     28 
2     21      1  !) 

19 
18 
17 
16 
15 

ItJ 
17 

18  ' 
19 

20 

2  53  S»i 
-3  4  3 
3  14  24 
.-I  024  4.2 
3  34  5« 

7      24      I  'J 
7     31      IS 
7      38        & 
7      44      51 
7      51      24 

.)      39         t 
9      40      51 
9     42     21 
9     43     4i 
9     44     5? 

:)      14     2fc 
>      1  0      54 
979 
9        3      13 

8      59        6 

o      25     4(! 
ti      18      lit 

!i      10      41' 

6      3    ir, 

5      55     S« 

212       t 
2        2      5.S 
1      53     Si 
1      44     1( 
1  '  34     54 

14 
13 
12 
11 
10 

21 
22 
23 
24 
25 

3  45  11 
3  55  21 
4  5  26 
4  25  26 
4  25  20 

7     57     45 
8       3     56 
S       9    57 
3     15     46 
S     21     24 

9     45     52 

9    46     38 
9     47     13 
9     47     36 
9    47     49 

8    54    50 
3     50    24 
8     45     48 
S    41       2 
8     36       6 

5    47    54 
5     40       4 
5     32       9 
5     24      9 
5     16      5 

1     25     31 

1     16      7 
1       6     41 
0    57     13 

10    47    44 

9 
8 
7 
6 
5 

26 
27 
28 
29 
30 

;4  35  6 
;4  44  42 
14  54  11 
5  3  33 
5  12  48 

3-   26     53 
3     32     11 
3     37     19 

13     42     18 

8    47      8 

|9     47     54 
9     47     46 
9     47     33 
9     47     14 
9     46     44 

IS     31       0 
8     25     44 
8    20     18 
8    .14     33 

8      8    59 

5      7     56 
4     59     42 
4    51     15 
4    43       2 
4    34    33 

JO     38     13 
;0    28     41 
0     19      8 
0      9     34 
003 

4 
3 
2 

0 

s? 

tfq 

11 

Signs 

10 

Signs 

9 

Signs 

8 
Signs 

7 
Signs 

6 

Signs 

a 
<£ 

Subtract 


192  Astronomical  Tables.  Sec.  14 

TABLE  X. 

The  third  equation  of  the  mean,  to  the  true  syzygy. 

AStGUMEJBT  r«-Su.ti*s  mean  anomaly — Moon's  equated  anomaly. 


B 

Si-ns. 

Sitjns. 

SLns. 

| 

'K 

3 

'   >!   ti  ,  Niii,;  ru1:!  . 

1    >i   M,  -...f.-       , 

i  S    n  ,  -a,     ,    . 

0^ 

3 

6  .Si  .'M--,  a  id. 

7  S5I..M1*,  adj. 

8     01       11--,     il    :'l. 

rt 

M         5? 

M        S 

M         S 

0 

2 
3 
4 
5 

0       0 
0       5 
0      10 
0      15 

0     20 
0     25 

2      22 
2     26 
2     30 
2      34 
2     33 
2     42 

4      12 
4      13 
4      18 
4      21 

4      24 
4     27 

30 
21 
28 
27 
26 
25 

'      6 
7 
3 
9 
10 

0     30 
0     35 
0     40 
0     45 
0      50 

2     46 
2     50 
2      £4 
2     58 
3       2 

4      30 
4     32 
4      34 

4     36  ' 
4     33 

-.4 
23 
22 
21 
20 

11 
12 
13 
14 
15 

0     o5 
0 
5 
10 
15 

3       6 
3      10 
3      14 
3     18 
3     22 

4     40 
4     42 
4     44 
4     46 
4     48 

19 

18 
17 
16 
15 

16 
17 

18 
19 
20 

20 
25 
SO 
35 
40 

3     26 
3     30 
3     34 
3     38 
3     42 

4     50 
4      51 

4     52 
4      53        . 
4      54 

14 
13 
12 
11 

10 

21 
22 
23 
24 
25 

45 
49 
52 
56 
2       0 

3     45 
3     43 
3     b\ 
3     54 
S     57 

4     55 
4     56 
4     57 
4      57 
4      57 

y 
8 
7 
6 
5 

26 
27 
?8 
29 
SO 

2       4 
2       9 
2      13 

2      18 

2      22 

4       0 
4       3 
4       6 
4       9 
4     12 

4     58 
4     58 
4     58 
4     58 
4      58 

4 
3 
2 
1 
0 

s? 

Si»ns. 

Signs. 

Si~ns. 

H 

Cfq 

5  Si  n<,  subtrar-t. 

4  £->i"ns,  si.b  rart.; 

3  Siens,  subtract. 

H 

S 

1  1  Sign*,  add. 

10  Signs,  add. 

y  Signs,  add. 

s 

Sec.  14 


Astronomical  Tables. 


193 


TABLE  XL 

The  fourth  equation  of  the  mean,   to  the  true  Syzygy. 


AR.G-UME1NT--T1IC  sun's  mean,  distance  from  flue  node* 

Add. 

V 

1 

» 

(A 

IT 
i 

2 
3 
4 
5 
~6~ 
7 
8 
9 
10 

ii" 

12 
13 
14 
15 

0    Signs. 
6    Signs. 

1    Sign. 

7   Signs 

2   Signs 
8   Signs 

1 

1JO 
29 
28 
27 
26 
25 
~24 
23 
22 
21 
20 

M.    S. 

M.    S. 

M.  S. 

0       0 
0       4 
0      7 
0     10 
0     13 
0     16 

1     22 
1     23 
1     24 
1     25 
1      26 
1     27 

1  22 
1  21 
1  20 
1  18 
1  16 
1  14 

0     20 
0     23 
0     26 

0     29 

0     32 

1     28 
1     29 
1     30 
1     31 
1     32 

1  12 
1  10 
1  8 
1  6 
1  3 

0     35 
0     38 
0     41 
0     44 
0     47 

1      33 
1     33 
1      34 
I     34 
1     34 

1  0 
0  57 
0  54 
0  51 

0  49 

19 
18 
17 
16 
15 

~14 
13 
12 
11 
10 

~~9 
8 
7 
6 

I 

~~4 
3 
2 
1 
0 

16 
17 
18 
19 
20 

2T 

22 
23 
24 
25 

0     50 
0     52 
0     54 
0     57 
1        0 

1      34 
1      34 
1      34 
1      33 
1      38 

0  45 
0  41 
0  37 
0  34 
0  31 

1       2 
1       5 
1        8 

I      10 
1      12 

1     32 
1     31 
1      30 
1     29 

1     28 

0  28 
0  25 
0  22 
0  19 
0  16 

26 
27 
28 
29 
30 

1      14 

1     16 

1      18 
I      20 
1      22 

1     27 
1     26 
1     25 
I      24 
1      22 

0  13 
0  10 
0  6 
0  3 
0  0 

5    Signs.  4    Signs.  !  3   Signs. 
11  Signs.  10Signs.|9    Signs. 

Subtract. 

194 


Astronomical    Tables. 
TABLE  XII. 


Stc.  14 


THE   SUN'S  MEAN  LONGITUDE— MOTION  AND  ANOMALY. 


Sun's'  mean 

I*ongittitlc»B«»Sun'8  mean 

anomaly* 

Years  beginning.    s 

D 

M 

S     ,    6 

D 

M 

8 

Old 

Style  1 

9 

7 

53 

10 

6 

23 

48 

1 

1! 

201 

9 

9 

23 

50 

6 

26 

67 

301 

9 

10 

9 

10 

6 

26 

1 

401 

9 

10 

54 

30 

6 

25 

5 

501 

9 

11 

39 

50 

6 

24 

9 

1001 

9 

15 

26 

30 

6 

19 

32 

1101 

9 

16 

11 

50 

6 

18 

36 

1201 

9 

16 

57' 

10 

6 

17 

40 

1301 

9 

17 

42 

30 

6 

16 

44 

1401 

9 

18 

27 

50 

6 

15 

49 

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9 

19 

13 

10 

6 

14 

53 

1601 

9 

19 

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30 

6 

13 

57 

1701 

9 

20 

43 

50 

6 

13 

1 

1801 

9 

21 

29 

10 

6 

12 

6 

N 

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9 

10 

37 

33 

6 

1 

8 

17 

1798 

9 

10 

23 

13 

6 

0 

52 

51 

1799 

9 

10 

8 

54 

6 

0 

37 

26 

1800 

9 

9 

54 

35 

6 

0 

22 

1 

1801 

9 

9 

40 

16 

6 

0 

6 

36 

1802 

9 

9 

25 

56 

5 

29 

51 

10 

1803 

9 

9 

11 

37 

5 

29 

35 

45 

1804 

9 

9 

56 

26 

6 

0 

19 

28 

1805 

9 

9 

42 

6 

6 

0 

4 

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1806 

9 

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27 

48 

5 

29 

48 

38 

1807 

9 

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29 

5 

29 

33 

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1808 

9 

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58 

17 

6 

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1809 

9 

9 

43 

57 

6 

0 

1 

31 

1810 

9 

9 

29 

37 

5 

29 

45 

57 

1811 

9 

9 

15 

17 

5 

29 

30 

32 

1812 

9 

10 

0 

5 

6 

0 

14 

15 

1813 

9 

9 

45 

45 

5 

29 

58 

49 

1814 

9 

9 

31 

25 

5 

29 

43 

26 

1815 

9 

9 

17 

5 

5 

29 

27 

58 

1816 

9 

10 

1 

53 

6 

0 

11 

41 

1817 

9 

9 

47 

33 

5 

29 

56 

15 

1818 

9 

9 

33 

13 

5 

29 

40 

50 

1819 

9 

9 

18 

53 

5 

29 

25 

24 

1820 

9 

10 

3 

41 

6 

0 

9 

7 

1821 

9 

9 

49 

22 

* 

3§ 

ill 

42 

Sec.  14  Astronomical  Tables. 

TABLE  XII.— CONTINUED. 

THE  SUN'S  MEAN  LONGITUDE— MOTION  AND  ANOMALY. 


195 


_  ?       Sun's  mean  Mt>tioii»«*3nu.*s  mean  anomaly* 

Years  complete.     s 

D 

31 

s      s 

D 

M 

1 

11 

29 

45 

40 

11 

29 

45 

2 

11 

29 

31 

21 

11 

29 

29 

3 

11 

29 

17 

20 

11 

29 

14 

4 

0 

0 

1 

50 

11 

29 

58 

5 

11 

29 

47 

31 

11 

29 

42 

A 

6 

11 

29 

33 

11 

11 

29 

27 

7 

11 

29 

18 

52 

11 

29 

11 

8 

0 

0 

3 

41 

11 

29 

55 

, 

9 

11 

29 

49 

21 

11 

29 

40 

10 

11 

29 

35 

2 

11 

29 

24 

11 

11 

,29 

20 

42 

11 

29 

9 

12 

0 

0 

5 

31 

11 

29 

53 

13 

11 

29 

51 

12 

11 

29 

37 

14 

11 

29 

36 

52 

11 

29 

22 

15 

11 

29 

22 

33 

11 

29 

7 

16 

0 

0 

7 

22 

11 

29 

50 

17 

11 

29 

53 

2 

11 

29 

35 

18 

11 

29 

38 

43 

11 

29 

20 

19 

11 

29 

24 

23 

11 

29 

4 

20 

0' 

0 

9 

12 

11 

29 

48 

40 

0 

0 

18 

24 

11 

29 

37 

60 

0 

0 

27 

36 

11 

29 

26 

80 

0 

0 

36 

48 

11 

29 

15 

100 

0 

0 

46 

0 

11 

29 

4 

• 

200 

0 

1 

32 

0 

11 

28 

8 

300 

0 

2 

18 

0 

11 

27 

12 

400 

0 

3 

4 

0 

11 

26 

16 

500 

0 

3 

50 

0 

11 

25 

21 

600 

0 

4 

32 

0 

11 

24 

25 

700 

0 

5 

17 

20 

11 

23 

29 

800 

0 

6 

2 

40 

11 

22 

331 

900 

0 

6 

48 

0 

11 

21 

37 

1000 

0 

7 

40 

0 

11 

20 

41 

2000 

0 

15 

20 

0 

11 

11 

22 

3030 

0 

22 

40 

0 

11 

2 

3 

4000 

1 

0 

13 

20 

10 

22 

44 

5000 

1 

7 

46 

40 

10 

13 

25 

6000 

1 

15 

30 

0 

10 

4 

I 

- 

196  Astronomical  Tables.  Sec.    14 

TABLE  XIL— CoKTimrED. 

THE  SUN'S  MEAN  LONGITUDE—MOTION  AND  ANOMALY. 


Sun's  mean  Motion—Sun's  mean  anomaly* 

MONTHS. 

s 

D 

M 

•  s            s 

D 

M 

• 

January, 

0 

0 

0 

0 

0 

0 

8 

February, 

1 

0 

33 

18 

1 

0 

3'd 

March, 

1 

28 

9 

11 

1 

28 

9 

April, 

2 

28 

42 

SO 

2 

28 

42 

May, 

3 

28 

16 

40 

3 

28 

17 

June,           1  4 

28 

49 

58 

4 

28 

50 

July, 

5 

28 

24 

8 

5 

28 

24 

August, 

6 

28 

57 

26 

6 

28 

57 

September, 

7 

29 

30 

44 

7 

29 

30 

•  "  • 

October, 

8 

29 

4 

54 

8 

29 

4 

November, 

9 

29 

38 

12 

9 

29 

37 

December. 

10 

29 

12 

22 

10 

29 

11 

Sun's  mean  motion  &>  anomaly—Sun's  mean  motion 

&  anomaly* 

DAYS 

S   D   M   S  IDAYS  |S   D   M    S 

1 

0   0  59   8 

17 

0  16  45  22 

2 

0   1  58  17 

18 

0  17  44  30 

3 

0   2  57  25 

19 

0  18  43  38 

4 

0   3  56  33 

20 

0  19  42  47 

5 

0   4  55  42 

21 

0  20  41  55 

6 

0   5  54  50 

22 

0  21  41   3 

7 

0   6  53  58 

23 

0  22  40  12 

8 

0   7  53   7 

24 

0  23  39  20 

9 

0   8  52  15 

25 

0  24  38  28 

10 

0   9  51  23 

26 

0  25  37  37 

1110  10  50  32 

27 

0  26  36  45 

12 

0  11  49  40 

28 

0  27  35  53 

13 

0  12  48  48 

29 

0  28  35   2 

14 

0  13  47  57 

30 

0  29  34  10 

15 

0  14  47   5 

31 

0  30  33  18 

16 

0  15  46  13 

See.  14  Astronomical  Tables.  197 

TABLE  XII.— CONCLUDED. 

THE  SUN'S  MEAN  LONGITUDE— MOTION  AND  ANOMALY. 
Sun's  mean,  motion  and  anomaly. 


H 
M 
S 
~I 

2 
3 

4 
5 

6 
7 
8 
9 
10 

Sun's  mean 
motion  and 
anomaly. 

Sun's  mean 
dist.  from 
the  Node. 

Sun's  mean 
motion  snd 
anomaly. 

Sun's  mean 
dist.  from 
the  Node. 

DM         S 
M       8     3dg 

s  3ds  4ths 

DM          8 
MS         T 
8       T         F 

H 
M 

S 

DM         S 
MS         T 

S      T          F 

DM          S 
MS         T 
S        T          F 

0      2     280      2    36 
0      4     560      5     12 
0      7    240      7    48 
0      9     510      0     23 
0     12     190     12    59 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

1     16    23 
1     18    51 
1     21     19 
1     23    47 
1     26     15 

1     20     30 
1     23      6 
1     25     42 
1     28     18 
1     30    54 

0     14     470     15     35 
0     17     150     18     11 
0     19    430     20     47 
0     22     110    23     23 

0    24    280     25    58 

1     28    42 
1     31     10 
1     33    38 
1     36      6 
1     38    34 

1     33    29 
1     36      5 
1     38     40 
1     41     16 
1     43    52 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
2f 
22 
23 
24 
25 

0     27      60     28     34 
0     29     340     31     10 
0     32      2;0     33    45 
0     34     300     36     21 
0     36     580    38    57 

,41 
42 
43 
44 
45 

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1     43     301     49     44 
1     45    571     51     39 
1     48    251     54     15 
1     50    531     55    51 

0     39     26 
0     41     53 
0     44     21 
0     46     49 
0     49     17 

0     41     33 
0    44      8 
0     46     44 
0    49     20 
0     51     56 

46 
47 
48 
49 
50 

1     53    21 
1     55    49 
1     58     17 

2       0     44 
2      3     12 

1     59     27 
22" 
2      4     39 
2      7     13 
2      9    50 

0    51     45 
0    54     13 
0    56     40 
0     59      8 
1       1     36 

0    54     32 
0    57      8 
0    59     43 
1       2     19 
1       4     55 

£1 
52 
53 
54 
55 
56 
57 
58 
59 
60 

2      5    40 
288 
2     10     36 
2     13      4 
2     15    32 

2     12    25 
2     15      2 
2     17     38 
2    20     14 
2    22    50 

26 
27 
28 
29 
30 

144 
1       6     32 
1       9      0 
1     11     23 
1     13    55 

1       7     31 
1     10      7 
1     12    43 
1     15     19 
1     17    55 

2     17    59 
2    20    27 
2    22    55 
2    25    23 

2    27    51 

2    25    26 
2    28      2 
2    30    38 
2     33    14 
2    35    50 

In  Leap- Year  after  February,  add  one  day,  and  one  day's  motion. 


198  Astronomical  Tables.  Sec.  M 

TABLE  XIII. 

Equation  of  the  Sun's  mean  distance  from  the  Node. 

ARGUMENT— Sun's  mean  anomaly. 


«> 

Subtract 

! 

« 

0S 

1*0 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 

0 

Signs. 

D   M 

1 

Sign 

D   M 

2 

.  Signs 

D   M 

3 

Signs 

D  M 

4 
Signs 

D  M 

5 

Signs. 

D   M 

0 

1 

2 
3 
4 
5 

0    0 

0   2 
0   4 
0   6 
0   9 
0  11 

I    2 

I   4 
I   6 
I   8 
I  10 
I  12 

1  47 
1  48 
1  49 
1  50 
1  51 
1  52 

2   5 
2   5 
2   5 
2   5 
2   5 
2   5 

I  50 
I  48 
I  47 
I  46 
I  45 
I  44 

1    4 

1   2 
1   0 

0  58 
0  56 
0  54 

6 

7 
8 
9 
10 

0  13 
0  15 
0  17 
0  19 
0  21 

I  14 
I  16 
I  17 
I  18 
I  19 

1  53 
1  54 
1  55 
1  56 
1  57 

2   5 
2   4 
2   4 
2   4 
2   4 

I  43 
I  41 
I  40 
T  38 
I  37 

0  52 
0  50 

0  48 
0  46 
0  44 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21" 
22 
23 
24 
25 

0  23 
0  25 
0  28 
0  30 
0  32 

I  21 
I  22 
I  24 
I  26 
I  27 

1  58 
1  58 
1  59 
2   0 
2   0 

2   3 
2   3 
2   3 
2   2 
2   2 

I  36 
I  34 
I  33 
I  31 
I  30 

0  42 
0  40 
0  37 
0  35 
0  33 

0  34 
0  36 
D  38 
0  40 
0  42 

I  28 
I  30 
I  31 
I  34 
I  35 

2   1 
2   1 
2   2 
2   2 
2   3 

2   1 
2   1 
2   0 
2   0 
1  59 

1  28 
I  27 
I  25 
I  24 
I  23 

0  31 
0  21 
0  27 
0  24 
0  22 

0  44 
0  46 
0  48 
0  50 
0  52 

1  36 
I  37 
I  39 

40 
41 

2   3, 
2   4 

S  i 

2   4 

1  59 
1  58 
1  57 
1  56 
1  55 

1  21 
19 
17 
15 
13 

0  20 
0  18 
0  16 
0  13 
0  11 

26 
27 

28 
29 
30 

0  54 
0  56 
0  58 
1   0 
1   2, 

43 
44 
45 
I  46 
I  47 

2   5 
2   5 
2   5 
2   5 
2   5 

54 
53 
52 
51 
50 

11 
9 
3 
6 
I   4 

0   9 
0   7 
0   5 
0   3 
0   0 

4 
3 
2 
1 
0 

J? 

11   10 

Signs  1  Signs  | 

9  j  8 
Signs]  Signs 

7 
Signs 

6 

Signs 

S? 

Add. 

See.  14  Astronomical  Tables, 

TABLE  XIV. 

THE  MOON'S  LATITUDE  IN  ECLIPSES. 


199 


ARGUMENT—  Moon's   Equated    Distance  from  the  Node. 

)   SIGNS  —  NORTH  ASCENDING. 

6  SIGNS  —  SOUTH  ASCENDING. 

Degrees. 

DM       s      [Degrees. 

0 

~0       0       0~ 

30 

1 

0       5     15 

29 

2 

0     10     30 

28 

3 

0     15     45 

27 

4 

0     20     59     '     26 

5 

0     26      13          25 

6 

0     31     26 

24 

.     7 

0     36     39 

23 

8 

0     41     51 

22 

9 

0     47     22 

2l 

- 

- 

10 

0     52      13 

20 

11 

0     57     23 

19 

12 

1       2     31 

8 

13 

1       7     38 

17 

14 

12     44 

16 

15 

17     49 

15 

16 

22     52 

14 

17 

27     53 

13 

18 

32     52 

12 

19 

37     49 

1  1 

FIVE  SIGNS. 

NORTH  DESCENDING. 

ELEVEN  SIGNS. 

SOUTH  DESCENDING. 

This  Table  shows  the  Moon's  true  Latitude  a  little  beyond 
the  utmost  limits  of  Eclipses. 


200 


Astronomical  Tables. 


Sec.  14 


TABLE  XIV. 

The  Moon's  horizontal  parallax,  with  the  semi-diameters,  and  the 
true  horary  motions  of  the  Sun  and  Moon,  to  every  sixth  degree  of  their 
mean  anomalies ;  the  quantities  for  the  intermediate  degrees,  being  ea- 
sily proportioned  by  sight. 


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Sec.  14 


Astronomical  Tables. 


201 


TABLE  XVI. 

MEAN  NEW  MOON,  &c.  IN  MARCH,  NEW  STYLE,  FROM  1800  TO 

1900  INCLUSIVE :  CALCULATED  FOR  THE  MERIDIAN 

OF  WASHINGTON.    76  DEGREES  AND  56 

MINUTES  WEST  LONGITUDE 

FROM  LONDON. 


Year 
of 

chrisi 

Mean    new 
Moon. 

Sun's    mean 
anomaly. 

Moon's  mean 
anomaly. 

Sun's  mean  dis- 
tance   from  the 
node- 

D     H.     M.    S. 

b      D.     M.  S. 

S     D.      M.     S. 

S    D.     M.      S. 

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27 

9 

9 

12 

ICT 

10 

2 

49  46 

1819 

-25   11   46  29 

8  23 

9 

39 

8 

14 

49 

18 

11 

11 

32  47 

182( 

13  20   35   » 

8   12 

25 

31 

6 

24 

37 

23 

11 

19 

35  34 

18-2! 

3    :>   *rj   4; 

3   l 

4! 

23 

5 

4 

25 

sa 

11 

2/ 

rS  21 

18  '2', 

-2-2   2  56  2i 

8  20 

3 

35 

4 

10 

2 

29 

1 

6 

22  22 

1828 

tl   11   44  59 

8   9 

19 

27 

2 

19 

50 

34 

1 

14 

25   9 

1824 

29   9L  17  39 

3  27 

41 

39 

1 

25 

27 

40 

2 

23 

9  10 

1825 

18   18   6   15 

8  16 

57 

31 

0 

5 

15 

45 

3 

1 

11  57 

202 


Astronomical  Tables. 


Sec.  14 


TABLE 


Year 
of 
Christ. 

Mean  new  Moon 
in  March. 

Sun's  mean 
Anomaly. 

Moon's  mean 
Anomaly. 

Sun's  mean  dis- 
tance from  the 
Node. 

I)    II   M    S 

S  D  M  S 

S    D    M    S 

S    D    M    S 

1826 

1327 
1823 
1829 

8   2  54  52 
•27   0  27  31 
15   9  16   8 
4  18   4  45 

23  15  37  24 

8  6  13  23 
8  24  35  35 
8  13  51  27 
8  3  7  19 
8  21  29  31 

10  15   3  50 
9  20  40  56 
8   0  29   1 
6  10  17   6 
5  15  54  12 

3   9  14  44 
4  17  58  45 
4  26   1  32 
5   4   4  19 
6  12  48  20 

1831 
1832 
1833 
1834 
1835 

13   0  28   1 
1   9  14  37 
20   6  47  17 
9  15  35  54 
28  13   8  33 

8  10  45  23 
8  "0  1  15 
8  18  23  27 
8  7  39  19 
8  26  1  31 

3  25  42  17 
2   5  30  22 
1  11   7  28 
11  20  55  33 
10  26  32  39 

6  20  51   7 
6  28  53  54 
8   7  37  55 
8  15  40  42 
9  24  24  43 

1836 
1837 
1838 
1839 
1840 

16  21  57  10 
6   6  45  46 
25   4  18  26 
14  13   7   2 
2  21  55  39 

8  15  17  23 
8  4  33  15 
8  22  55  27 
8  '12  11  19 
3  1  27  11 

9   6  20  44 
7  16   8  49 
6  21  45  55 
5   1  34   0 
3  11  22   5 

10   2  27  30 
10  10  30  17 
11  19  14  18 
11  27  17   5 
0   5  19  52 

1841 
1842 
1843 
1844 
1845 

21  19  28  19 
11   4  16  55 
30   1  49  35 
18  10  38  12 

7  19  26  48 

8  19  49  23 
8  9  5  15 
8  27  27  27 
8  16  43  19 
8  5  59  11 

2  16  59  11 
0  26  47  16 
11  32  24  22 
10  12  12  27 
8  22   0  32 

1  14   3  53 
1  22   6  40 
3   0  50  41 
3   8  53  28 
3  16  56  15 

1846 
1847 
1848 
1849 

1850 

26  16  59  28 
16   1  48   5 
4  10  36  41 
23   8   9  21 
12  16  57  57 

8  24  21  23 
8  13  37  15 
8  2  53  7 
8  21  15  19 
8  10  31  11 

7  27  37  38 
6   7  25  43 
4  17  13  48 
3  22  50  54 
2   2  38  59 

4  25  40  16 
5   3  43   3 

5  11  45  50 
6  20  29  51 
6  28  32  38 

iSji 
1852 
1853 

1854 
1855 

2   1  46  33 
19  23  19  13 

9   8   7  49 
28   5  40  29 
17  14  29   5 

7  29  47  3 
8  18  9  15 
,8  7  25  7 
8  25  47  19 
8  15  3  11 

0  12  27   4 
11  18   4  10 
9  27  52  15 
9   3  29  21; 
7  13  17  26; 

7   6  35  25 
8  15  19  28 
8  23  22  13 
10   2   6  14 
lO  10   9   1 

14 


Tables. 


203 


TABLE  XVI—  CONTINUED. 


Year 
of 
Christ. 

Mean  new  Moon 
in  March. 

Sun's  mean 
Anomaly. 

Moon's  mean 
Anomaly. 

Snn's  mean  dis- 
tance from  the 
Node. 

D    II    M    S 

S  D  M  S 

S   D    M    S 

S    D    MS 

1856 
1857 
1858 
1859 
1860 

5  23  17  42 
24  20  50  22 
14   5  28  58 
3  14  27  35 
21  12   0  15 

8  4  19  2 
8  22  41  15 
S  11  57  7 
3  1  12  59 
3  19  35  11 

5  23   5  31 
4  28  42  37 
3   8  30  42 
1  18  18  47 
0  23  55  53 

10  18  11  48 
11  26  55  49 
0   4  58  36 
0  13   1  23 
1  21  45  24 

1861 
1862 
1863 
1864 
1865 

10  20  48  51 
29  18  21  31 
19   3  10   7 
7  11  58  44 
26   9  31  24 

8  8  51  3 
8  27  13  15 
8  16  29  7 
3  5  45  0 
8  24  7  12 

11   3  43  58 
10   9  21   4 
8*19   9   9 
6  28  57  14 
6   4  34  20 

1  29  48  11 
3   8  32  12 
3  16  34  59 
3  24  37  46 
5   3  21  47 

1866 
1867 
1868 
1869 

1870 

15  18  20   0 
5   3   8  37 
23   0  41  16 
12   9  29  53 
1  18  18  30 

8  13  23  4 
8  2  38  56 
8  21  1  S 
8  10  17  0 
7  29  32  52 

4  14  22  25 
2  24  10  30 
1  29  47  36 
0   9  35  41 
10  19  23  46 

5  11  24  34 
5  19  27  21 
6  28  11  22 
7   6  14   9 
7  14  16  56 

1871 

1872 
1873 
1874 
1875 

20  15  51   9 
9   0  39  46 
27  22  12  26 
17   7   1   2 
6  15  49  39 

8  17  55  4 
8  7  10  56 
8  25  33  8 
8  14  49  0 
8  4  4  52 

9  25   0   5 
8   4  48  10 
7  10  25  16 
5  20  13  21 
4   0   1  26 

8  23   0  57 
9   1   3  44 
10   9  47  45 
10  17  50  32 
10  25  53  19 

1876 

1877 
1878 
1879 
1880 

24  13  22  18 
13  22  10  55 
3   6  59  32 
22   4  32  11 
10  .13  20  48 

8  22  27  4 
8  11  42  56 
8  0  58  48 
8  19  21  0 
8  8  36  52 

3   5  38  32| 
1  15  26  37 
11  25  14  42 

11   0  51  48 
9  10  39  53 

0   4  37  20 
0  12  40   7 
0  20  42  54 
1  29  28  55 
2   7  29  42 

1881 
1882 
1883 
1884 
1885 

29  10  53  28 
18  19  42'   4 
8   4  30  41 
26   2   3  20 
15  10  51  57 

8  26  59  5 
8  16  14  57 
8  5  30  49 
8  23  53  1 
8  13  8  53 

8  16  16  59 
6'  26   5   4 
5   5  53   9' 
4  11  30  15! 
2  21  18  20j 

3  16  13  43! 

3  9,-i  16  30  i 
4   2f  19  17 
5  11   3  18  i 
5  19   6   5: 

204 


Astronomical  Tables. 


Sec.  14 


TABLE  XVI— CONCLUDED. 


y«w 

of 
Ciirisi 

Mean  new  Moon 
in  March. 

Sun's  mean 
Anomaly. 

Moon's  mean 
Anomaly. 

Sun's  mean  dis- 
tance from  the 
Node. 

D   H    M   -8 

s  n  M  s 

S    D    M    S 

S    D    M    S 

1886 
1837 
1888 
1839 
1890 

4  19  40  33 
23  17  13  13 
12   2   1  50 
1  10  50  26 
20   8  23   6 

8  2  24  45 
3  20  46  57 
3  10  2  49 
7  29  18  41 
8  17  40  53 

1   1   6  25 
0   6  43  31 
10  16  31  36 
8  26  19  41 
8   1  56  47 

5  27   «  52 
7   5  52  53 
7   13  55  40 
7  21  58  27 
9   0  42  28 

1891 
1892 
1893 
1894 
1895 

9  17  11  43 
27  14  44  22 
16  23  32  59 
6   8  21  35 
25   5  54  15 

3  6  56  45 
8  25  18  57 
8  14  34  49 
8  3  50  41 
3  22  12  5. 

6  11  44  52 
5  17  21  58 
3  27  10   3 
2   6  53   8 
1  12  34  14 

9   8  -io  15 
10  17   2  24 
10  25  32   3' 
11   3  34  50, 
0  11  18  51 

1896 
1897 
1898 
1899 
1903 

13  14  42  5'2 
2  23  31  28 
21  21   4   9 
11   5  52  4' 
30   3  25  24 

8  11  28  43 
3  0  44  37 
3  19  6  4« 
8  8  22  41 
8  26  44  53 

11  22  22  lt> 
10   2  10  24 
9   7  46  33 
7  17  34  35 
|  6  23  13  41 

0  19  21  33 
0  27  24  25 
2   6   8  26 
2  14  11  13 
|  3  22  53  14 

The  year  1900,  will  not  be  Leap-Year,  the  differ- 
ence then  will  be  13  days,  between  the  Old  and 
New  Style. 


Sec.  14 


Astronomical  Tables. 
TABLE  XVII. 


205 


A  concise  EQUATION  TABLE,  adapted  to  the  second  year  after  Leap- 
Year,  within  one  minute  of  the  truth,  for  every  year,  (excepting  the 
second)  showing  to  tha>  nearest  full  minute,  how  much  a  Clock  should  be 
faster  or  slower  than  the  Sun. 


?       t 

3               «£ 

rf* 

I    S 

ff 

I  ! 

if 

f    1 

ll 

5.  S' 

s-     *5 

f  § 

S-      M 

a  § 

f  I 

S^           v. 

ft    3 

i* 

*  =' 

p 

5 

p: 

CO 

"  r- 

January,   1 
3 

4 
5' 

April,  1 
4 

|| 

August,  1 

i^ 

October,^'. 
November,!-  i 

ib 
Id 

5 

6J 

7 

2( 

3r- 

2( 

I4C 

7 

10 

1  -r 

')  «- 

24 

132 

£ 

8  F 

15 

Oa 

28 

II 

2? 

12* 

15 
15 

9rt 

10: 

IS 

_  

31 

0? 

30 
December,  g 

11  SE. 
10* 

If 

n? 

24 

2' 

September,  ? 

1 

5 

<}  t 

2: 

l'2  = 

3 

35 

A 

ty  **' 

*J 

Or 

13   T 

May,  1  3 

1  ^ 

4  ^ 

f 

3f 

r 

3 

1    1 

Q  x. 

H 

A    7" 

i  t 

6- 

February,!*" 

June,  5 

V2  5 

16 

05 

•      1:? 

•        *  21 

i 

If 

6§ 

J 

4  'f 

2* 

lo 

o' 

21 

7  ^ 

IF 

3  = 

March,  * 

12 
11 

2: 

;' 

2* 
I 

1 
I 

to 

0 

2f 

2: 

•22 

S'' 
Octobe;,  ^ 

10  L 

11  « 

24 

|J 

q 

i  ^ 

3' 

f. 

12  r 

2(J 

1  - 

2 

7 

Jil|y>  o 

4? 

re 

13  = 

2f 

2  r-. 

i 

8 

ii 

14 

14 

3' 

3  t 

2 

5 

ti 

6^ 

I! 

15 

- 

$^>This  Table  is  near  enough  to  the  truth,  for  regulating  common  Clocks 
and  Watches,  and  was  for  that  purpose  calculated  by  Mr.  Smeatou, 


SECTION  FIFTEENTH. 


EXAMPLE  I. 


Required  the  true  time  of  new  Mooon  in  July,  1832, 
and  also  whether  there  were  an  Eclipse  of  the  Sun 
or  not. 


Mc;in  nov/ 


arch  1832 
5  lunaiions, 

D 

147 

it 

=.— 

15 

M 

"14" 

40 

15'; 

from  Table  7 

149 

0 
1 

54 
46 

52 

°, 

;ime  once  eq'td. 

Table  9th. 

26 

23 
2 

8 
12 

43 
8 

Table  10th. 

26 

20 

56 
3 

35 
38 

Table  llth. 

* 

26 

20 

62 

57 

49 

26 

20 

53 
6 

46 

Eq'tn.  of  the 
|  Sun's  centre. 

26 

20 

47 

46 

' 
.:a!y. 

Moon's  mean 
Anomaly. 

rSun's  mean  dist 
from  the  Node. 

j;I  '       S 

4  25  &1     3? 

S       D       M    ,    VS 

27T"30"~22 
49       5-2 

S  D  .  M  S 

6~28~53  54 
5  3  21  10 

0  25  38    5"2 

6  14     35     24 

0  2  14  04 

6  13  55       6 

4  11  a?     4J 

40     IS 

• 

Argument  4th. 

Argument  2. 

Argument  3d. 

Equal  to  the  27th  day  of  July,  8  hours,  47  minutes,  and  46  seconds  in 
the  morning,  at  WASHINGTON  ;  the  true  time  of  new  Moon.  The  Sun 
being  then  only  two  degrees  and  14  minutes  from  the  Moon's  ascending 
mode,  w*i  consequently  eclipsed. 


Sec.  15 


Examples. 


207 


EXAMPLE  II. 

Required  the  true  time  of  new  Moon  in  May,  1836, 
and  whether  there  will  be  an  Eclipse  of  the  Sun 
or  not. 


Mean  new  Moon 
in  March,  1836. 

Sun's  mean 
Anomaly. 

Moon's  mean 
Anomaly. 

Sun's  m.  dist. 
from  the  node. 

D 

16 

H 

M 

s 

S        D 

M 

s 

S       I>       31       S 

S       D 

JVI 

s  ' 

IQQfi 

21 
1 

57 

28 

10    8     15 
6    1     28 

17 
12 

23 
39 

9     6  20  44 

1   21   :  8     1 

10     2 
2     1 

27 
20 

30 

28 

Table  3d 

Table  7th. 

14 

- 

23 
3 

2~ 

5 

~w 

15 

IF 
21 
57 
1 

16 

'  24 
40 
50 
44 
20 

10     13 

30 
6 

E 

r; 

10  27  53  45 
\     8     2 

0     3 

47 

JTable  9th. 
Table  10th. 

11     14 

23 

i 

10  29.    6  47 

Table  llth. 

14 

20 

59 

4 
13 

Table  17lh. 

14 

20 

59 
4 

17 

1 

114 

21 

3 

18 

Equal  to  the  loth  day  of  May,  9  hours,  3  minutes, 
and  18  seconds ;  true  time  of  new  Moon  at  WASHING- 
TON. The  Sun  being  then  only  13  degrees  and  48 
minutes  from  the  Moon's  Node,  the  Sun  will  conse- 
quently be  visibly  eclipsed. 


208 


Examples. 


Sec.  15 


EXAMPLE  III. 


Required  the  true  time  of  New  Moon,  in  Decem- 
ber, in  the  year  1850;  and  whether  there  will  be  an 
Eclipse  at  that  time  or  not. 


Mean  New  Moon 
in  March,  1350. 

Sun's  mean 
Anomaly. 

Moon's  mean 
Anomaly. 

Sun's  mean 
distance  from 
'     the  Node. 

D         11         MS 

S        D       M         S 

8         D       M         S 

S        D      M          S 

12     16     57    57 
265     18     36    27 

8     10    31     11 

8     21     56     54 

2       2     38    59 
7     22     21       4 

6     f.8     32      38 
9626 

3     11     34     24 

5      2    28      5 

9     25       0       3 

4       4     34      44 

I     53      3 

9    24     15     25 

44     38 

.  ', 

3       9     36     21 

7       8     12     40 

9     24     15     25 

9     11     13 

•••  "  •  — 

3       0    25       8 
2     54 

3      0    28      2 

1     27 

3       0     26     35 
9    42 

3       0     36       7 

True  time  of  new  Moon  in  December,  1850,  will 
be  the  3 J.  36;;: — 17s,  afternoon.  The  Mm  will  then 
be  more  thin  56  degrees  from  the  node,  and  conse- 
quently there  can  be  no  Eclipse  at  that  time. 


Sec.  15. 


Examples. 


209 


EXAMPLE  I. 


Required  the  true  time  of  full  moon  in  July  1833, 
and  whether  or  not  there  will  be  an  eclipse  of  the 
moon,  at  that  time. 


Mean  new 
Moon  in  March 

D    H.    M.   S. 

Sun's  mean 
anomaly 

S  D.   M.    S. 

Moon's  mean 
anomaly 

S  D.   M.    S. 

Sun's    mean 
distance  from 
the  node 

S    D.  M.    S. 

20 

!  ) 

6 
14 

18 

47      17 
12       9 

22       2 

8      IS 
2     27 
0      14 

23  27 
18  5S 
33  1  (i 

i     11 
2     17 

6     12 

i    zo 
27       1 
54     30 

8       7 
3       2 
0     15 

37 
0 

20 

5") 
42 
7 

i 

15 

il      28 

1        7 

•  )        vj 
'0      11 

1  o  i5o 
28  31 

.0      11 

28     oy 

25 

1  1     24 

58 

4-J 

L 

10 

7 

20     21 

42       8 

1     18     47       1  10     11 

in  the  afternoon. 

28     34, 

i 

7 

33     13 
3     38 

1 

7 

34     35 
16 

1 

7 

34     19 
3     22 

I 

7 

37     41 

=To  the  first  day  of  July,  1833  the  true  time  of  full 
moon  in  the  longitude  of  Washington,  at  7  hours  37 
minutes  and  41"  seconds  in  the  afternoon,  the  sun,  be- 
ing then  within  five  degrees  at  a  mean  rate  from  the 
Moon's  node,  consequently  the  Moon  will  then  be  e- 
clipsed. 


210 


Examples. 


Sec.  15. 


Required  the  true  time  of  full  Moon  in  April,  in  the 
year  1836  at  Rochester;  and  also,  whether  there  will 
be  an  eclipse  of  the  Moon,  or  not. 

The  true  time  of  full  Moon,  in  April,  in  the  year 
1836,  will  be  on  the  first  day,  5  hours  19  minutes  and 
53  seconds  in  the  afternoon,  in  the  longitude  of  Ro- 
chester ;  the  sun  will  then  be  more  than  40  degrees 
from  the  Moon's  node,  and  consequently  there  will 
be  no  eclipse  on  that  day. 

EXAMPLE  VII. 

Required  the  true  lime  of  full  Moon,  in  September  in  the  year  1848,  in 
the  longitude  of  Utica  ;  and  whether  there  will  be  an  eclipse  at  that  time. 


Meantime  of  full 
Moon  in  March 

Sun's  mean 
anomaly. 

Moon's    mean 
anomaly. 

Sun's  mean  dis- 
distance  from  the 
node. 

1)    H. 

M. 

S. 

S     D, 

M. 

8. 

S 

D. 

M.       S.  S 

D.        M. 

S. 

4  10 
177    4 

14  18 

36 
24 

22 

4i 
18 
2 

8       3 
5     24 

0     14 

37 
33 

7 
56 

10 

4 
5 
6 

17 
4 
12 

13       48 
54         3 
54       30 

(3 
0 

11         45 
4           1 

15         20 

53 

24 

7 

12     9 
3 

23 

57 

1 

O 

2     12 
4       3 

4 
32 

13 
30 

4 

5 

1 

2       21 
29       61 

p 

1           7 

21 

12     5 
.T 

25 

48 

h9 
40 

10       8     31 

True  time  at 
True  time  at 

43J4 

Lyons 
Utica. 

a 

32      30 

12   13 

14 
3 

46 

12  13 

18 

25 
5 

12  13 

18 
4 

30 
8 

12  13 

22 
6 

38 

12  J3 

28 

38 

The  Moon  will  be  full  in  the  year  1848,  on  the  13th  day  of  September,  at 
1  o'clock  and  28  minutes  in  the  morning,  in  the  longitude  of  Utica.  the 
Sun,  then  will  be  only  one  degree  and  seven  minutes  from  the  Moon's  node; 
the  Moon  there f9re,  will  be  eclipsed  at  that  time. 


Sec.  15  Examples.  211 

To  calculate  the  true  time  of  any  new,  or  full  Moon, 
and  consequently  Eclipses,  in  any  given  year  and 
month,  between  the  commencement  of  the  Christian 
Era,  and  that  of  the  18th  Century. 
Find  a  year  of  the  same  number  in  the  18th  Centu- 
ry, with  that  of  the  year  in  the  proposed  Century  from 
Table  First;  and  take  out  the  mean  time  of  New 
Moon  in  March,  Old  Style,  for  that  year ;  with  the 
mean  anomalies  of  the  Sun  and  Moon,  and  the  Sun's 
mean  distance  from  the  node  at  that  time,  as  before 
instructed.  Take  as  many  complete  Centuries  of 
years  from  Table  Second  as  when  subtracted  from  the 
year  of  the  18th  Century,  the  remainder  will  answer 
to  the  given  year,  with  the  anomalies,  and  Sun's  dis- 
tance from  the  node  ;  subtract  these  from  those  of  the 
18th  Century,  and  the  remainder  will  be  the  mean 
time  of  new  Moon  in  March,  with  the  anomalies,  §*c. 
for  the  proposed  year ;  then  proceed,  in  all  respects, 
for  the  true  time  of  new  or  full  Moon,  as  shown  in  the 
Precepts,  or  former  Examples. 

If  the  day's  annexed  to  these  Centuries,  exceed  the 
number  of  daysfr  om  the  beginning  of  March,  taken  out 
in  the  18th  Century,  subtract  a  lunation,  and  its  anom- 
alies, fyc.  from  Table  3d,  to  the  time,  and  anoma- 
lies of  new  Moon  in  March,  and  then  proceed  as  above 
stated — this  circumstance  happens  in  Example  Fifth. 


212 


Examples. 
EXAMPLE  IV. 


Sec.  15 


Required  the  true  time  of  New  Moon  in   June,  in 
the  year  of  Christ,  36,  at  the  City  of  JERUSALEM. 


BY  THE  PRECEPTS 

Mean  N.  MOOI. 
in  March. 

Sun's  mean 
Anomaiy. 

Moon  -;  n  ean 
Anomaly. 

Sun's  mean 
dist.  from 
the  Node. 

D         H      M        fe 

D      D        AI        S 

S  *  D         M          S 

S        D        M        S 

8      D    M        S 

March,  1736. 
Add  one  lunation. 

0     18     54       £ 
.9     12     44       '. 

6     11     52     22 
0    29       6     H 

8     20     58     49 
0     i5     49       0 

5  12  24    27 
1     9  40    14 

'jast  n.  moon,  March  173\ 
Subtract,  1700  vear?. 

,0       7     38       5 
9     19     11     25 

3     10     58     41 
3     19     ;8     48 

1      1U     47     4o 
1     22     10     37 

b   13  34    41 
6  14  31      7 

.nean  n.moon  March  in  £C 
Add  three  lunations. 

l\     15    26    4( 
38     14     12       J 

0     20     59     53 
2     27  .  18     58 

11     24     17     K 
2     17     27       1 

21  29  3     34 
3     2     0    4$ 

By  Table  Fourth= 

110 

June 

18      5    38    4£ 

3     18     IS     51 

2     11     44     13 

J     1     4    16 

First  equation  

3    54    4i 

Arg  for  1st  eqt. 

1     30     55 

18       1     39     C< 
9     24     1$ 

3     ly     18     51 

2     10     13     18 

2     10     13     It 
Arg.  2:1.  eqlion. 

3     r    4    16 

Ar?.4  h  eqi'u 

Third  equation.-  

18     11       3    21 

2    54 

1       8      5     33 

Ar£.  3d.  'qt'n 

18     11       0     26 
3 

True  time  of  New  Mooi. 
at  LONDON  

18     11       0     24 
2    -20 

True  time  al  JERUSALEM 

18     13    20     24 

The  true  time  of  New  Moon  in  June,  in  the  year  of 
our  Lord,  36,  on  the  19th  day,  at  one  hour,  20  minutes 
and  24  seconds,  in  the  morning. 

The  mean  distance  of  the  Sun  being  3  signs,  1  de- 
gree, 4  minutes,  and  16  seconds  from  the  Moon's  as- 
cending node,  consequently  there  was  no  Eclipse 
at  that  time. 


Sec.  15  Precepts  and  Examples.  213 

To  calculate  the  true  time  of  new,  or  full  Moon,  and 
also  to  know  whether  there  will  be  an  Eclipse  at  the 
time,  in  any  given  year  and  month  before  the  Christian 
Era.  Find  a  year  in  (he  18th  Century  from  Table  1st. 
which  being  added  to  the  given  number  of  years  before 
Christ  diminished  b>  one,  shall  make  a  number  of  com- 
plete Centuries.  Find  ihis  number  of  Centuries  in  Ta- 
ble seeond,  and  subtract  tfte  time,  anomalies  and  dis- 
tanccs  from  the  node  belonging  to  it,  from  those  of  the 
mean  new  Moon  in  March,  the*  above  found  year,  in 
the  18th  Century,  and  the  remainder  will  c'enote  the 
time,  and  anomalies,  £LC.  of  the  mean  new  Moon  in 
M .irch,  the  given  year,,  before  Christ, ;  then  for  the  true 
time  thereof  in  any  month  of  that  year,  proceed  us  be- 
fore directed, 


214 


Precepts  and  Examples. 
EXAMPLE  V. 


Sec.  15 


Required  the  true  time  of  new  Moon  in  May,  Old 
Style,  the  year  before  Christ,  585,  at  ALEXANDRIA,  in 
EGYPT.  The  year  584,  added  to  1716,  make  2300,  or 
23  Centuries. 


Mean  N.  Moon 
in  March. 

Sun's  mean 
Anomaly. 

Moon's  mean 
Anomaly. 

Sun's  mean  dis- 
tance from  the 
Node. 

D          H      M        S 

S        D        M        S 

S         D        M          S 

S         D        M           S 

March,  1716  

11     17    33    29 

8    22    50    39 

4      4     14       2 

4     27     17        5 

March,  2000 
Do.       300.     .  . 

27     18      9     1. 
13      0    32    37 

0      8    50      0 
0     10      3      0 

0     15    42      0 
Ilfi      fi      f) 

6     27     45        0 

1       9R      99      H   0 

Subtract  1  lunation  .  . 

40     18    52    56  1 
29     12    44      3 

0     18     53      0 
•0    29      6     19 

2       1     48       0 
0    25    49      0 

8     26     07        0 
1       0    40      14 

New  Moon,  2300,.... 

11       5    58    53 

11     J9    46    41 

1       5     59      0 

7     25     26      46 

Which  sub't.      m!716 
March,  B.C.  585... 
Add  3  lunations  .... 

0     11     34    36 
88     14    12      9 

9      3    03    58 

2    27     18    58 

2    28     15      2 
2     17    27       1 

9       1     50      19 
3      2      0      42 

New  Moon  March  58,. 
First  equation  

28       1     46    45 
1     37 

0      0    22    56 
Argt.  1st.  eqi'n. 

5     15    42      3 

46 

0      3    51      01 

28       1     45    08 

0      0    22    56 
5     15    41     17 

5     15     41     17 

Argt,  2d.  eq'tn. 

0      3    51      01 

Second  equation  

28       1     45    08 
2     15       1 

6     14    41     3i) 

Arg't.  3d.  eqt'n. 

5     15     41     17 

0      3    51        1 

Third  equation  

28       4    00       9 
1       9 

6     14    41     39 

5     15    41     17 

0      3    51        1 

Argt.  4th.  eqt'n. 

Fourth  equation... 

28       4    01     18 
12 

Clock  slower.... 

28      4       1     30 
3 

Time  at  LONDON  
Difference  of  longitude. 

28      4      4    30 
2      2 

May 

28      6      6    30 

The  true  time  of  new  Moon  at  ALEXANDRIA  in  May, 
585  years  before  Christ,  was  on  the  28th  day,  6  hours, 
6  minutes  and  30  seconds,  afternoon.  The  Sun  being 
then  only  three  degrees  and  51  minutes  from  the 
Moon's  ascending  node  ;  was  consequently  eclipsed. 

The  above  Eclipse  was  central,  and  total  in  NORTH 
AMERICA  at  eleven  o'clock  in  the  morning;  it  also  pas- 
sed centrally  over  the  south  parts  of  FRANCE  and  ITALY. 
The  duration  of  total  darkness  being  about  3  minutes. 


Sec.  15 


Precepts  and  Examples. 
EXAMPLE  VIII. 


215 


Required  the  true  time  of  full  Moon,  at  ALEXANDBIA 
in  EGYPT,  in  September,  Old  Style,  in  the  year  201  be- 
fore the  Christian  Era.  200  years  added  to  1800, 
make  2000,  or  20  Centuries. 


By  the  PRECEPTS 

Mean  n.  Moon 
in  March. 

Sun's  mean 
Anomaly. 

Moon's  mean 
Anomaly. 

Sun's  dis.frm 
the  Moon's 
ascnd'g  node 

D      H    M      S 

S    D     M      S 

S     D     M      S 

S      D    M    S 

March,  1800. 
Add  1  lunation. 

13    0  22  17 
29  12  44     3 

8  23  19  55 
0  29    6  19 

10    7  52  S3 
0  25  49    0 

11     8  58  24 
1     0  40  14 

From  the  same, 
Subtract  2003  yr's. 

42  13  06  20 
27  18     9  19 

9  22  26  14 
0    8  50    0 

11     3  41  36 
0  15  42    0 

0     4  38  38 
6  27  45    0 

m.  n.  moon  b.C.201 
Add  6  lunations. 

14  18  57  01 
177     4  24  18 

9  13  36  14 
5  24  37  56 

10  17  59  36 
5    4  54    3 

5    6  53  38 
6     4     1  24 

191— 

n.m.Sept.b.C.201 
Add  \  lunation, 

7  23  21  19 
14  18  22    2 

3    8  14  10 
0  14  33  10 

3  22  53  39 
6  12  54  30 

11  10  55    2 

0  15  20    7 

full  moon  Sep.  201 
First  equation. 

22  17  43  21 
3  52    6 

3  22  47  20 
Arg.  1st.  eqt. 

10    5  48     9 
1  28  14 

11  26  15     9 

Time  once  eqt'd. 
Second  equation. 

22  13  52  15 

8  25    4 

3  22  47  20 
10    4  19  55 

10    4  19  55 
Arg.  2d.  eqt. 

11  26  15     9 

Time  twice  eqt'd. 
Third  equation. 

22     5  26  11 

58 

5  18  27  25 
Arg.  3d.  eqt. 

10     4  19  55 

11  26  15    9 
Arg.  4th  eqt- 

99      ^  9^   1  3 

12 

true  T.at  LONDON 

22    5  25     1 

add  for  diffoflong 

2    2 

Add  S.  Clock. 

22    7  27     1 
7  33 

True  Time  at 
ALEXANDRIA. 

22    7  34  34 

The  true  time  of  full  Moon,  at  ALEXANDRIA  in  EGYPT 
in  the  year  before  Christ,  201,  in  September,  was  on  the 
22d  day,  7  hours,  34  minutes,  and  34  seconds,  the  actu- 
al time  of  opposition.  The  Sun  being  within  three  de- 


216  Examples  and  Precepts  Sec.  15 

grees  and  45  minutes  of  the  Moon's  ascending  node, 
consequently  the  Moon  was  visibly  eclipsed  at  that  time 
at  ALEXANDRIA. 

To  calculate  the  true  time  of  new  or  full  Moon,  and 
Eclipses  in  any  given  year  ;  and  month  after  the  18th 
Century. 

Find  a  year  of  the  same  number  in  the  18th  centu- 
ry with  that  of  the  year  proposed,  and  take  out  the 
mean  time,  and  anomalies  Sfc,  of  new  moon  in  March, 
old  style,  from  table  first  for  that  year. 

Take  so  many  years  from  table  second,  as  when 
•added  to  the  above  mentioned  year  in  the  18th  century 
will  answer  to  the  given  year  in  which  the  new,  or 
full  moon  is  required;  and  take  out  the  first  new  Moon 
with  its  anomalies  &,c  for  these  complete  centuries. 
Add  all  these  together,  and  then  proceed  as  before 
directed,  to  reduce  the  mean  to  the  true  syzygy.  It 
is  however  necessary  to  remember,  to  subtract  a  lu- 
nation with  its  anomalies,  when  the  above  said  .addi- 
tion carries  the  new  Moon  beyond  the  31st  day  of 
March,  as  in  the  following  example. 


Sec.  15 


Examples  and  Precepts. 
EXAMPLE  IX. 


217 


Required  the  true  time  of  New  Moon  in  July,  old 
style,  2180  at  Washington. 


Four  centuries  added'*  to  178O  make  2180. 

New  Moon 
in  March 

Sun's    mean 
anomalies 

Moon's  mean 
anomalies. 

Sun's  dis- 
tance from 
the  node. 

By  the  precepts. 

D.    H.    M.    S 

S.    D.    M.    S. 

S.     D.    M.    S. 

S.     D.    M.    S. 

March  1780 
add  400  years 

23  23     1  44 
17    8  43  29 

9    4  18  13 
0  13  24    0 

1  21    7  47 
10     1  28    0 

10  18  21    1 
6  17  49    0 

Subtract  1  lunation 

41     7  45  13 
29  12  44     3 

9  17  42  13 
0  29     6  19 

11  22  35  47 
0  25  49    0 

5    6  10     1 
1     0  49  14 

Mean  time  new 
Moon  March  2180 
add  4  lunations 

11  19     1  10 
118    2  56  12 
129 

8  18  35  54 
3  26  25  17 

10  26  46  47 
3  13  16    2 

4    5  29  47 
4    2  40  56 

tf.  M.  July  2180 
First  equation 

7  21  57  22 
1     3  39 

0  15     1  11 
Arg.  Isteqt. 

2  10    2  49 
24  12 

8    8  10  43 

Time  once  equated 
Second    equation 

7  20  53  43 
9  24    8 

0  15     1   11 
2     9  38  37 

2    9  38  37 

Arg.  2nd  eqt 

8    8  10  43 
Arg.  4th  eqt 

i    j.i     „  t. 

Third  equation, 

8    6  17  51 
3  56 

10     5  22  34 
Arg.  3d  eqt. 

\ 
i    t            M 

Fourth  equation 

8     6  21  47 
1     8 

H  f                              11 

F.  Clock 

8    6  22  55 
4  30 

T.  time  at  London 
Difference  of  long. 

8    6  IS  25 
5    8 

True  time  at  Wash- 
ington  the  present 
Capitol  of  the  TJ.  S. 

8     1  10  25 

The  true  of  time  New  Moon,  old  style,  will  then  be 
on  the  8th  day  of  July,  1  hour  10  minutes  and  25  sec- 
onds after  noon ;  or  the  22d  day,  at  the  same  hour 
minute  and  second  new  style. 


218 


Examples  and  Precepts 
EXAMPLE  IX. 


Sec.  15 


Required  the  Sun's  true  place,  March  20th,  1764  O. 
Stvle,  at  22  hours,  30  minutes,  25  seconds  past  noon. 
In  common  reckoning,  March  21st,  at  10  hours  30  min- 
utes and  25  seconds  in  the  morning. 


Sun's  Long.  Sun's  Anmly 

SDMSJSDMS 


To  the  Radical  year  after  Christ, 1701    9  20  43  50    6  13     1     0 

Add  complete  years, 60;  0     0  27  1211  29  26    0 

3'11  29  17     0  11  29  14    0 


March, 

Bissextile  days, 20 

Hours, 22 

-,-  Minutes, 30 

Seconds, 25 


Sun's  mean  place  at  the  given  time, 

Add  equation  of  the  Sun's  centre,  from  table  6? 

Sun's  true  place, 

That  is  ARIES,  12deg's.  10  minutes,  12sec'ds. 


1  28  9  111  1  28  9  0 
0  20  41  55  0  20  41  5 
54  13 
1  14 


0  10  14  36 


54  13 
1  14 
1 


9     1  27  23 


0  12  10  12  with  which 
enter  table  6., 


Sec.  15 


Precepts  and  Examples. 
EXAMPLE  XI. 


219 


Required  the  Sun's  true  place,October  23,  (X  Style, 
at  16  hours,  57  minutes  past  noon,  in  the  4008th  year 
before  Christ  L  which  was  the  4007th  year  before  the 
year  of  his^  birth,  and  the  year  of  the  Julian  period, 
706.  This' is  supposed  by  some  to  be  the  very  instant 
of  the  Creation. 


BY  THE  PRECEPTS. 

Sun's  Long 

Sun's  anomy 

S      D     M    S 

S      D     M     S 

9     7  53  10 

1     7  46  40 

6  28  48    0 
10  13  25    0 

Remains  for  a  new  Radix,  
To  which  add  C  900 

8    0     6  30 
0    6  48    0 
0    0  36  16 
0    0    5  26 
8  29    4  54 
0  22  40  12 
39  26 
2  20 

8  15  23    0 
11  21  37    0 
11  29  15    0 
11  29  15    0 
8  29    4    0 
0  22  40  12 
39  26 
2  20 

Complete  years,  <    80 
C    12 

Hours,  

Minutes,  

Sun's  mean  place  at  the  given  time,  

6034 
3     4 

5  28  3?  58 

Subtract  equation  of  the  Sun's  centre,  

Argt.  eqt'n. 
Sun's  centre 

6000 

Which  was  just  entering  the  Sign  LI^RA. 

220     Concerning  Eclipses  of  tin  Sun  and  Moon.  Sec.  15 
Concerning  Eclipses  of  the  £um  & 


To  find  the  Sun's  true  distance  from  the  Moon's  as- 
cending node,  at  the  time  of  any  given  new  or  full 
Moon,  and  consequent!}  to  know  whether  there  be 
an  Eclipse  at  that  time  or  not. 

The  Sun's  mean'distance  from  the  Moon's  ascending 
node,  is  the  Argument  for  finding  the  Moon's  fourth 
equation  in  the  syzygies,  and  therefore  it  is  taken  into 
all  the  foregoing  Examples,  in  finding  the  true  times 
thereof.  Thus  aj  the  time  of  mean  new  Moon  in  March> 
1764,  Old  Style,  or  April  in  the  new,  the  Sun's  mean 
distance  from  the  ascending  node,  is  0  signs,  35  min- 
utes, 2  seconds.  [See  Table  First.]  The  descending 
node  is  opposite  to  the  ascending  one,  and  consequent- 
ly are  exactly  6  signs  distant  from  each  other.  When 
the  Sun  is  within  17  degrees  of  either  of  the  nodes  at 
the  time  of  new  Moon  ;  he  will  be  eclipsed  at  that  time, 
as  before  stated,  and  at  the  time  of  full  Moon,  if  ihe  Sun 
be  within  12  degrees  of  either  node,  she  will  be  eclipsed. 
Thus  we  find  from  Table  First,  that  there  was  an 
Eclipse  of  the  Sun,  at  the  time  of  new  Moon,  April 
1st.  at  30  minutes,  25  seconds  after  10  in  the  morning 
at  LONDON,  New  Style,  when  the  old  is  reduced  to  the 
new,  and  the  mean  time  reduced  to  the  true. 

It  will  be  found  by  the  Precepts,  that  the  true  time 
of  that  new  Moon  is  50  minutes,  46  seconds  later,  than 
the  mean  time,  and  therefore  we  must  add  the  Sun's 
motion  from  the  node  during  that  interval  to  the  abov,e 
mean  distance  Os.  6d.  35m,  2s.  which  motion  is  found 


Sec.  15  Elements  for  Solar  Eclipses.  221 

in  Table  Twelfth,  for  50  minutes  and  46  seconds  to  be 
2  minutes,  12  seconds,  and  to  this  apply  the  equation 
of  the  Sun's  mean  distance  from  the  node  in  Table  13th, 
which  at  the  mean  time  of  new  Moon,  April  1st.  1764, 
is  9  signs,  1  degree,  26  minutes,  and  20  seconds,  and 
we  shall  have  the  Sun's  true  distance  from  the  node  at 
the  true  time  of  new  Moon,  as  follows  : 

Sun  from  node, 
s  D   M    s 

At  the  mean  time  of  N.Moon  in  April,  1764,  0  5  35  2 
Sun's  motion  from  node  for  50  minutes,  2  10 

For  46  seconds,  2 

Sun's  mean  dist  from  node  at  true  N.  Moon,  0  5  87  14 
Equation  from  mean  dist  from  node,  add,  250 
Sun's  true  dist.  from  the  ascending  node,  0  7  42  14 
Which  being  far  within  the  above  named  limits  of  17 
degrees,  the  Sun  was  at  that  time  eclipsed.  The  man- 
ner of  projecting  this  or  any  other  Eclipse,  either  of  the 
Sun  or  Moon,  will  now  be  shown. 


SECTION  SIXTEENTH. 


To  Project  an  Eclipse  of  the  Sun. 

To  project  an  Eclipse  of  the  Sun,  we  must   from   the 

Tables  find  the  ten  following  Elements  : 

1st.  The  true  time  of  conjunction  of  the  Sun  and 
Moon,  and 

2d.  The  semi-diameter  of  the  earth's  disk,  as  seen 
from  the  Moon,  at  the  true  time  of  conjunction  ;  which 
is  equal  to  the  Moon's  horizontal  parallax. 

3d.  The  Sun's  distance  from  the  solstitial  colure?  to 
which  he  is  then  nearest. 

4th.  The  Sun's  declination.     . 

5th.  The  angle  of  the  Moon's  visible  path  with  the 
ecliptic. 

6th.  The  Moon's  latitude. 

7th.     The  Moon's  true  horary  motion  from  the  Sun. 

8th.  The  Sun's  semi-diameter, 

9th.  The  Moon's  semi-diameter. 

10th.  The  semi-diameter  of  the  penmb  ra. 


Sec.  16    Elements  for  Protracting  Solar  Eclipses.    223 
EXAMPLE  XII. 


Required  the  true  time  of  New  Moon  at  LONDON,  in 
April,  1764,  New  Style,  and  also  whether  there  were 
an  Eclipse  of  the  Sun  or  not  at  that  time  ;  arid  likewise 
the  elements  necessary  for  its  protraction,  if  there  were 
at  that  time  an  Eclipse. 


By  the  Precepts. 

mean  time  of 
New     Moon 
in  March. 

Sun's    mean 
anomaly. 

Moons  mean 
anomaly. 

Sun's    mean 
dist'.from 
the   node. 

D    H     M     S 

S     D     M      S 

S     D     M      S 

S      D     M      S 

March  1764 
Add  1  lunation 

2     8  55  36 
29  12  44     3 

8     2  20    0 
0  29    6  19 

10  13  35  21 
0  25  49    0 

11     4  54  48 
1     0  40  14 

Mean  New  Moon 
First  equation 

31  21  39  39 
4  10  40 

9     1  26  19 
Arg  1st  eqt'n 

11     9  24  21 
1  34  57 

0     5  35     2 

Second  equation. 

32     1  50  19 
3  24  49 

9     1  26  19 
11  10  59  18 

11  10  59  18 
Arg  2nd  eqt' 

Third  equation 

31  22  25  30 
4  37 

9  20  27     1 
'Arg  3d  eqt'n 

11  10  59  18 

0    5  35     2 

Arg  4th  eqt' 

31  22  30    7 

18 

Sun  from 
node 

True  New  Moon 
Equation  of  days 

31  22  30  25 
3  48 

0    5  35    2 

31  22  26  35 

The  true  time  is  April  1st,  10  hours,  26  minutes,  35 
seconds  in  the  morning,  tabular  time.  The  mean  dis- 
tance of  the  Sun  at  that  time,  being  only  5  degrees,  35 
minutes  and  2  seconds  past  the  ascending  node,  the 
Sun  was  at  that  time  eclipsed.  Now  proceed  to  find 
the  elements,  necessary  for  its  protraction.  The  true 
time  being  found  as  above. 

To  find  the  Moon's  horizontal  parallax,  or  semi-di- 
ameter of  the  Earth's  disk  as  seen  from  the  Moon. 


224  Declination  of  Solar  Eclipses.  Sec.  16 

Enter  Table  15th  with  the  signs  and  degrees  of  the 
Moon's  anomaly,  (making  proportion  because  the 
anomaly  is  in  the  Table  calculated  only  to  every  6th 
degree,)  and  fyom  it  take  out  the  Moon's  horizontal 
parallax,  which  for  the  above  time  is  54  minutes,  and 
53  seconds,  answering  to  the  anomaly  of  11s.  9d. 
24m.  21  seconds. 

To  find  the  Sun's  distance  from  the  nearest  solstice, 
namely,  the  beginning  of  Cancer,  which  is  3  signs,  or 
90  degrees  from  the  beginning  of  Aries.  It  appears 
from  Example  1st.  for  calculating  the  Sun's  true  place, 
the  calculation  being  made  for  the  same  time,  that  the 
Sun's  longitude  from  the  beginning  of  Aries,  was  then 
Os.  12d.  10m.  12  seconds,  that  is,  the  Sun's  place  was 
then  in  Aries,  12  degrees,  10m.  12  seconds,  therefore 
from  s  D  M  s 

3000 

Subtract  the  Sun's  longitude  or  place.  12  10   12 

Remains  Sun's  distance  from  the  solstice,  2  17  49  48 
Which  is  equal  to  77  degrees,  49  minutes,  48  seconds, 
each  sign  containing  30  degrees. 

To  find  the  Sun's  declination,  enter  Table  5th  with 
the  signs  and  degrees  of  the  Sun's  true  place,  anomaly 
10s.  2d.  and  making  proportions  for  the  10m.  12  sec- 
onds, take  out  the  Sun's  declination,  answering  to  his 
true  place,  and  it  will  be  found  to  be  4  degrees^49  min- 
utes north. 

To  find  the  Moon's  latitude,  this  depends  on  her  true 
distance  from  her  ascending  node,  which  is  the  same  as 
the  Sun's  true  distance  from  it  at  the  time  of  new  Moon, 


Sec.  16  Declination  of  Solar  Eclipses.  225 

and  is  thereby  found  in  Table  14th.  But  we  have  al- 
ready found,  [see  Example.]  by  calculation  the  Sun's 
true  place,  thatat  the  true  time  of  new  Moon  in  April, 
1764,  the  Sun's  equated  distance  from  the  node  was 
Os.  7d.  42m.  14s.  therefore  enter  Table  14th  with  the 
above  equated  distance,  (making  proportions  for  the 
minutes  and  seconds,)  her  true  latitude  will  be  found 
to  be  40  minutes  and  18  seconds  north  ascending. 

To  find  the  Moon's  horary  motion  from  the  Sun, 
with  the  Moon's  anomaly,  namely,  11s.  9d.  24m.  21s. 
enter  Table  15th,  and  take  out  the  Moon's  horary  mo- 
tion, which,  by  making  proportions  in  that  Table,  will 
be  found  30  minutes,  22  seconds. 

Then  with  the  Sun's  anomaly,  namely,  9s.  Id.  26m. 
19s.  (in  the  present  case,)  take  out  his  horary  motion, 
2  minutes  and  28  seconds  from  the  same  Table  ;  sub- 
tract the  latter  from  the  former,  and  the  remainder  will 
be  the  Moon's  horary  motion  from  the  Sun  ;  namely,  27 
minutes  and  54  seconds. 

To  find  the  angle  of  the  Moon's  visible  path  with  the 
ecliptic.  This  in  the  projection  of  Eclipses,  may  be  al- 
ways rated  at  5  degrees,  and  35  minutes  without  any 
sensible  error. 

To  find  the  semi-diameters  of  the  Sun  and  Moon. 

These  are  found  iu  the  same  Table,  [15]  and  by  the 
same  Argument,  as  their  horary  motions.  In  the  pres- 
ent case,  the  Sun's  anomaly  gives  his  semi-diameter, 
16  miuutes,  and  6  seconds,  and  the  Moon's  anomaly 
gives  her  diameter  14rn.  and  27  seconds. 


226  Declination  of  Solar  Eclipses:  Sec,  1  fr 

To  find  the  semi-diameter  of  the  penumbra.      Add  the 
,  Sun's  semi-diameter  to  the  Moon's,  and  their  sum  will 
be  the  semi-diameter  of  the  penumbra  ;  equal  to  31 
minutes  and  3  seconds. 

Collect  these  elements  together,  that  they  may  the 
more  readily  be  found  when  they  are  wanted  in  the 
construction  of  this  Eclipse.  Thus-  - 

D    H  M    s 

1st.  The  true  time  of  new  Moon  in  April,    1   10  30  25 

D     M     s 

2d.  Semi-diameter  of  the  Earth's  disk,  0  54  53 

3d.  Sun's  distance  from  the  nearest  solistic,  77  49  48 
4th.  Sun's  declination  north,  4  49  0 

5th.  Moon's  latitude,  north  ascending,  0  40  18 

6th.  Moon's  horary  motion  from  the  Sun,  0  27  54 
7th.  Angle  of  Moon's  visible  path  with  ecliptic.  5  35  0 
8th.  Sun's  semi-diameter.  016  6 

9th.  Moon's  semi- diameter.  14  57 

10th.  Semi-diameter  of  the  penumbra.  51     3 

To  project  an  Eclipse  of  the  Sun  Geometrically  : — 
Make  a  scale  of  any  convenient  length,  A.  C.  and  di- 
vide it  into  as  many  equal  parts,  as  the  Earth's  semi- 
disk  contains  minutes  of  a  degree  5  which,  at  the  time 
of  the  Eclipse  in  April,  1764,  was  54  minutes  and  53 
seconds ;  then  with  the  whole  length  of  the  scale  as  a 
radius,  describe  the  semicircle  A.  M.  B.  upon  the  cen- 
tre C.  which  semi-circle  will  represent  the  northern 
half  of  the  Earth's  enlightened  disk,  as  seen  from  the 
Sun, 


&ec.  16  Declination  of  Solar  Eclipses.  227 

Upon  the  centre  C.  rises  the  straight  line,  Ch.  per- 
pendicular to  the  diameter,  A.  C.  B.  then  will  A.  C.  B. 
be  a  part  of  the  ecliptic,  and  C.  H.  its  axis. 

Being  provided  with  a  good  sector,  open  it  to  the  ra 
dius  C.  A.  in  the  line  of  Chords,  and  taking  from  thence 
the  chord  of  23  degrees  and  28  minutes  in  your  com- 
passes, set  it  off  both  ways  from  H.  to  g.  fy  g.  to  h.  in 
the  periphery  of  the  semi-disk,  and  draw  the  straight 
line  g.  V.  h.  in  which  the  north  pole  of  the  disk  will 
be  always  found. 

When  the  Sun  is  in  Aries,  Taurus,  Gemini,  Cancer, 
Leo  and  Virgo,  the  north  pole  of  the  Earth  is  enlight- 
ened by  the  Sun,  but  while  the  Sun  is  in  the  other  six 
signs,  the  south  pole  is  in  the  dark. 

When  the  Sun  is  in  Capricorn,  Aquarius,  Pisces, 
Aries,  Taurus  and  Gemini,  the  northern  half  of  the 
Earth's  axis,  C.  XII.  P.  lieslo  the  right  hand  of  the 
axis  of  the  ecliptic,  as  seen  from  the  Sun  ;  and  to  the 
left  hand,  whilst  the  Sun  is  in  the  other  6  signs. 

Open  *  the  sector,  till  the  radius,  [or  distance  of  the 
two  90s]  of  the  signs  be  equal  to  the  length  of  V.  h. 
and  take  the  sign  of  the  Sun's  distance  from  the  solis- 
tice  77  degrees,  49  minutes,  and  48  seconds  in  your 


*  To  persons  acquainted  with  Trigonometry,  the  angle  contained  be- 
tween the  Earth's  axis,  and  that  of  the  ecliptic,  may  be  found  more  ac- 
curately by  calculation. 

RULE. — As  Radius  is  to  the  sine  of  the  Sun's  distance  from  the  sol- 
stice, so  is  the  tangent  of  the  distance  of  the  poles,  (23  degrees  and  28 
minutes,)  to  the  tangent  of  the  angle  contained  by  the  axis.  Then  set  off 
the  chord  of  the  angle,  from  H.  to  h.  and  join  C.'H.  which  will  cut  F.  G. 
in  P.  the  place  of  the  north  pole. 


S28  Declination  of  Solar  Eclipses,  Sec.  16 

compasses  from  the  line  of  sines,  and  set  off  that  dis- 
tance from  V.  to  P.  in  the  line  of  g.  V.  h.  because  the 
Earth's  axis  lies  to  the  right  hand  of  the  axis  of  the 
ecliptic  in  this  case,  [the  Sun  being  in  Aries,]  and  draw 
the  straight  line  C.  XII.  P.  for  the  Earth's  axis,  of 
which  P.  is  the  north  pole.  If  the  Earth's  axis  had 
lain  to  the  left  hand  from  the  axis  of  the  ecliptic,  the 
distance  V.  P.  would  have  been  set  off  from  V. 
towards  g. 

To  draw  the  parallel  of  latitude  of  any  given  place, 
as  suppose  for  LONDON  in  this  case,  or  the  path  of  that 
place  on  the  Earth's  enlightened  disk,  as  seen  from  the 
Sun,  from  Sim-rise  to  Sun-set  take  the  following 
method. 

Subtract  the  latitude  of  LONDON  in  this  case,  51  de- 
grees and  30  minutes,from  90  degrees,  and  the  remain- 
der 38  degrees  and  30  minutes  will  be  the  co-latitude, 
which  take  in  your  compasses  from  the  line  of  chords, 
making  C.  A,  or  C.  B.  the  radius,  and  set  it  from  h.  to 
the  place  where  the  Earth's  axis  meets  the  periphery 
of  the  disk  to  VI.  and  VI.  and  draw  the  occult  or  dot- 
ted line  VI.  K.  VI.  then  from  the  points  where  this 
line  meets  the  Earth's  disk  set  off  the  chord  of  the 
Sun's  declination,  [4  degrees  and  49  minutes,]  to  A. 
and  F.  and  to  E.  and  G.  and  connect  these  points  by 
the  two  occult  lines.  F.  XII.  G.  and  A.  L.  E. 

Bisect  L.  K.  XII.  in  K.  and  through  the  point  K. 
draw  the  black  line  VI.  K.  VI.  then  making  C.  B.  the 
radius  of  a  line  of  sines  on  the  sector,  take  the  co -lati- 
tude of  LONDON,  (38  and  J  degrees,)  from  the  sines  in 


Sec.  16  Declination  of  Solar  Eclipses.  229 

your  compasses,  and  set  it  both  ways  from  K.  to  VI. 
and  VI.  These  hours  will  be  just  in  the  edge  of  the 
disk  at  the  equinoxes,  but  at  no  other  time  in  the  whole 
year.  With  the  extent  K.  VI.  taken  into  your  com- 
passes, set  one  foot  in  K.  in  the  black  line  below  the 
occult  one  as  a  centre,  and  with  the  other  foot  describe 
the  semi-circle  VI.  7.  8.  9.  1(X  &c.  and  divide  it  into 
12  equal  parts  ;  then  from  these  points  of  division,draw 
the  occult  7.  p.  8.  0.  9.  n.  parallel  to  the  Earth's  axis, 

C.  XII.  P. 

With  the  small  extent  K.  XII.  as  a  radius,  describe 
the  quadrantal  arc  XII.  f.  and  divide  it  into  six  equal 
parts,  as  XII.  a.  a.  b.  be.  cd.  de.  ef.  and  through  the 
division  points  a.  b.  c.  d.  e.  draw  the  occult  lines  VII. 
e.  V.VIII.  d.  IV.  IXC.  III.  X.  b.  II.  and  XI.  a.  I.  all 
parallel  to  VI.  K.  VI.  and  meeting  the  former  occult 
lines  7.  p.  8.  0.  #c.  in  the  points  VII.  VIII.  IX.  X. 
XL  V.  IV.  III.  II.  and  I.  which  points  will  mark  the 
several  situations  of  LONDON  on  the  Earth's  disk  at 
these  hours  respectively,  as  seen  from  the  Sun,  and  the 
elliptic  curve  VI.  VII.  VIII.  fyc.  being  drawn  through 
these  points,  will  represent  the  parallel  of  latitude,  or 
path  of  LONDON  on  the  disk  as  seen  from  the  Sun  from 
its  rising  to  its  setting. 

If  the  Sun's  declination  had  been  south,  the  diurnal 
path  of  LONDON  would  have  been  on  the  upper  side  of 
the  line  VI.  K.  VI.  and  would  have  touched  the  line 

D.  L.  E.  in  L.     It  is  necessary  to  divide  the  hourly 
spaces  into  quarters,  and  if  possible  into  minutes  also. 


230  Declination  of  Solar  Eclipses.  Sec.  16 

Make  C.  B.  the  radius  of  a  line  of  chords  on  the  sec- 
tor, and  taking  therefrom  the  chord  of  5  degrees  and 
35  minutes,  (the  angle  of  the  Moon's  visible  path  with 
the  ecliptic  ;)  set  it  off  from  H.  to  M.  on  the  left  hand 
of  C.  H.  (the  axis  of  the  ecliptic,)  because  the  Moon's 
latitude  in  this  case  is  north  ascending.  Then  draw 
C.  M.  for  the  axis  of  the  Moon's  orbit,  and  bisect  the 
angle  M.  C.  H.  by  the  right  line  C.  Z.  If  the  Moon's 
latitude  had  been  north  descending,  the  axis  of  her  or- 
bit would  have  been  on  the  right  hand  from  the  axte 
of  the  ecliptic. 

The  axis  of  the  Moon's  orbit  lies  the  same  way  when 
her  latitude  is  south  ascending,  as  when  it  is  north  as- 
cending, and  the  same  way  when  south  descending,  as 
when  north  descending. 

Take  the  Moon's  latitude  (40  minutes  and  18  sec- 
onds,) from  the  scale  C.  A.  in  your  compasses,  and  set 
it  from  i.  to  x.  in  the  bissecting  line  C.  Z.  making  i.  x. 
parallel  to  C.  y.  and  through  x.  at  right  angles,  to  the 
Moon's  orbit,  (C.  M.)  draw  the  straight  line  N,  w.  x. 
y.  s.  for  the  path  of  the  penumbra's  centre  over  the 
Earth's  disk. 

The  point  w.  in  the  axis  of  the  Moon's  orbit,  is,  that, 
where  the  penumbra's  centre  approaches  nearest  to 
the  centre  of  the  Earth's  disk,  and  consequently  is  the 
middle  of  the  general  Eclipse,  The  point  x.  is  where 
the  conjunction  of  the  Sun  and  Moon  falls,  according 
to  equal  time,  as  calculated  by  the  Tables,  and  the 
point  y.  is  the  ecliptical  conjunction  of  the  Sun  and 
Moon. 


Sec.  1 6  Declination  of  Solar  Eclipses.  23 1 

Take  the  Moon's  true  horary  motion  from  the  Sun, 
•  (27  minutes  and  54  seconds,)  in  your  compasses,  from 
the  scale  C.  A.  (every  division  of  which  is  a  minute  of 
a  degree,)  and  with  that  extent,  make  marks  along  the 
path  of  the  penumbra's  centre,  ^nd  divide  each  space 
from  mark  to  niark,  into  60  equal  parts,  or  horary 
minutes  by  dots,  and  set  the  hours  to  every  60th  -min- 
ute in  such  manner,  that  the  dot  signifying  the  instant 
of  new  Moon  by  the  Tables,  may  fall  into  the  point  x. 
halfway  between  the  axis  of  the  Moon's  orbit,  and  the 
axis  of  the  ecliptic  ;  and  then  the  remaining  dots  will 
be  the  points  on  the  Earth's  disk,  where  the  penum- 
bra's centre  is  at  the  instants  denoted  by  them,  in  its 
transit  over  the  Earth. 

Apply  one  side  of  a  square  to  the  line  of  the  penum- 
bra's path,  and  move  the  square  backwards  and  for- 
wards, until  the  other  side  of  it  cuts  the  same  hour  and 
minute,  (as  at  m.  and  m.)  both  in  the  path  of  the  pe- 
numbra's centre,  and  the  particular  minute,  or  instant 
which  the  square  cuts  at  the  same  time  in  both  paths, 
will  be  the  instant  of  the  visible  conjunction  of  the  Sun 
and  Moon,  or  the  greatest  obscuration  of  the  Sun  at 
the  place  for  which  the  construction  is  made,  (namely, 
LONDON-  in  this  Example,)  and  this  instant  is  at  47 
minutes  and  29  seconds  past  10  o'clock  in  the  morning, 
which  is  17  minutes,  5  seconds  later  than  the  tabular 
time  of  true  conjunction. 

Take  the  Sun's  semi-diameter,  (16  minutes  and  six 
seconds,)  in  your  compasses,  from  the  scale  C,  A.  and 
setting  one  foot  in  the  path  of  LONDON,  at  m.  viz.  at 


232  Declination  of  Solar  Eclipses.  Sec.  16 

47  minutes  and  thirty  seconds  past  10,  with  the  other 
foot  describe  the  circle  U.  Y.  which  will  represent  the 
Sun's  disk  as  seen  from  LONDON  at  the  greatest  ob- 
scuration. 

Then  take  the  Moon's  semi-diameter,  fourteen  min- 
utes and  57  seconds  in  your  compasses  from  the  same 
scale,  and  setting  one  foot  in  the  path  of  the  penumbra's 
centre  at  m.  in  the  47  and  J  minute  after  10,  with  the 
other  foot  describe  the  circle  T.  Y.  for  the  Moon's  disk, 
as  seen  from  LONDON,  at  the  time  when  the  Eclipse  is 
at  the  greatest,  and  the  portion  of  the  Sun's  disk, 
which  is  hidden,  or  cut  off  by  the  disk  of  the  Moon, 
will  show  the  quantity  of  the  Eclipse  at  that  time, 
which  quantity  may  be  measured  on  a  line  equal 
to  the  Sun's  diameter,  and  divide  it  into  12  equal  parts 
for  digits,  which,  in  this  Example,is  nearly  eleven  digits. 
This  Eclipse  was  annular  at  PARIS. 

Lastly,  take  the  semi-diameter  of  the  penumbra,  3 1 
minutes  and  3  seconds  from  the  scale  A.  C.  in  your 
compasses,  and  setting  one  foot  in  the  line  of  the  penum- 
bra's path,  on  the  left  hand,  from  the  axis  of  the  eclip- 
tic, direct  the  other  foot  towards  the  path  of  LONDON, 
and  carry  that  extent  backwards  and  forwards,  until 
both  the  points  of  the  compasses  fall  into  the  same  in- 
stants in  both  the  paths,  and  these  instants  will  denote 
the  time  when  the  Eclipse  begins  at  LONDON,  Proceed 
in  the  same  manner  on  the  right  hand  of  the  axis  of  the 
ecliptic,  and  where  the  points  of  the  compasses  fall  into 
the  same  instants  in  both  the  paths,  they  will  show  at 
what  time  the  Eclipse  ends  at  LONDON. 


Sec.  16  Defoliation  of  Solar  Eclipses.  233 


d  i&mter  of  th&TenWftCbrti     .<• 
kor&ry  matter*  fre**  <*«  Su* 


Sun'*    semi  dt&mtter 
semi  diameter 


Sec.  16  Defoliation  of  Solar  Eclipses.  235 

According  to  this  construction,  this  Eclipse  began 
at  20  minutes  after  9  in  the  morning,  at  LONDON,  at  the 
points  N.  and  O.  47  minutes  and  30  seconds  after  10, 
at  the  points  m.  and  m.  for  the  time  of  the  greatest  ob- 
scuration, and  18  minutes  after  12,  atE.  and  S.  for  the 
time  when  the  Eclipse  ends. 

In  this  construction,  it  is  supposed  that  the  angles 
under  which  the  Moon's  disk  is  seen  during  the  whole 
time  of  the  Eclipse,  continues  invariably  the  same,  and 
that  the  Moon's  motion  is  uniform,  and  rectilinear  du- 
ring that  time.  But  these  suppositions  do  not  exactly 
agree  with  the  truth  and  therefore  supposing  the  ele- 
ments given  by  the  Tables  to  be  accurate,  yet  the 
times  and  phases  of  the  Eclipse  deduced  from  its  con- 
struction, will  not  answer  to  exactly  what  passes  in  the 
Heavens,  but  may  be  at  least  two  or  three  minutes 
wrong,  though  the  work  may  be  done  with  the  great- 
est care  and  attention. 

The  paths  also,  of  all  places  of  considerable  latitudes 
are  nearer  the  centre  of  the  Earth's  disk  as  seen  from 
the  Sun,  than  those  constructions  make  them ;  be- 
cause the  disk  is  projected  as  if  the  Earth  were  a  per- 
fect sphere,  although  it  is  known  to  be  a  spheroid. 

The  Moon's  shadow  will  consequently  go  farther  north- 
ward in  all  places  of  northern  latitude,  and  farther 
southward  in  all  places  of  southern  latitude,  than  can 
be  shown  by  any  projection. 


SECTION  SEVENTEENTH, 


The  Projection  of  JLunar  JEcltpses. 


WHEN  the  Moon  is  within  12  degrees  of  either  of 
her  nodes,  at  the  time  when  she  is  full,  she  will  be 
eclipsed,  otherwise  not,  as  before  stated, 

Required  the  true  time  of  full  Moon,  at  LONDON,  in 
May,  1762,  New  Style,  and  also  whether  there  were 
an  Eclipse  of  the  Moon  at  that  time  or  not. 

It  will  be  found  by  the  Precepts,  that  at  the  true 
time  of  full  Moon  in  May,  1762,  the  Sun's  mean  dis- 
tance from  the  ascending  node  was  only  4  degrees,  49 
minutes  and  36  seconds,  and  the  Moon  being  then  op- 
posite to  the  Sun,  must  have  been  just  as  near  her  de- 
scending node,  and  was  therefore  eclipsed.  The  ele- 
ments for  the  construction  of  Lunar  Eclipses  are  eight 
in  number,  as  follows  : 


Sec.  17  Delini ation  of  Lunar  Eclipses.  237 

1st,  The  true  time  of  full  Moon. 

2d.  The  Moon's  horizontal  parallax. 

3d.  The  Sun's  semi-diameter. 

4th.  The  Moon's  semi-diameter. 

5th.  The  semi-diameter  of  the  Earth's  shadow  at 
the  Moon. 

6th.  The  Moon's  latitude. 

7th.  The  angle  of  the  Moon's  visible  path  with  the 
ecliptic. 

8th.  The  Moon's  true  horary  motion  from  the  Sun. 

To  find  the  true  time  of  full  Moon,  proceed  as  di- 
rected in  the  Precepts,  and  the  true  time  of  full  Moon 
in  May,  1762,  will  be  found  on  the  8th  day,  at  50  min- 
utes, and  50  seconds  past  3  o'clock  in  the  morning. 

To  find  the  Moon's  horizontal  parallax,  enter  Table 
15th  with  the  Moon's  mean  anomaly,  ^at  the  time  of 
the  above  full  Moon,)  namely,  9s.  2d.  42m.  42  seconds, 
and  with  it  take  out  her  horizontal  parallax,  which,  by 
making  the  requisite  proportions  will  be  found  to  be 
57  minutes  and  23  seconds. 

To  find  the  semi-diameters  of  the  Sun  and  Moon, 
enter  Table  15th,  with  their  respective  anomalies,  the 
Sun's  being  10s.  7d.  27m.  45  seconds,  and  the  Moon's 
9s.  2d.  42m.  42  seconds,  (in  this  case,)  and  with  these 
take  out  their  respective  semi-diameters,  the  Sun's  15 
minutes  and  56  seconds,  and  the  Moon's  15  minutes 
and  38  seconds. 

To  find  the  semi-diameter  of  the  Earth's  shadow  at 
the  Moon,  add  the  Sun's  horizontal  parallax,  (which 
is  always  9  seconds,)  to  the  Moon's  which  in  the  pres- 


238  Deliniation  of  Lunar  Eclipses.  £cc.  17 

ent  case  is  57  minutes  and  23  seconds,  the  Sun  will  be 
57  minutes  and  32  seconds  ;  from  which  subtract  the 
Sun's  semi-diameter,  15  minutes  and  56  seconds,  and 
there  will  remain  41  minutes  and  36  seconds  for  the 
semi-diameter  of  that  part  of  the  Earth's  shadow, 
which  the  Moon  then  passes  through. 

To  find  the  Moon's  latitude.  Find  the  Sun's  true 
distance  from  the  Moon's  ascending  node,  (as  already 
taught,)  in  the  first  Example  for  finding  the  Sun's  true 
place,  at  the  true  time  of  full  Moon,  and  this  distance 
increased  by  6  signs,  will  be  the  Moon's  true  distance 
from  the  same  node,  and  consequently  the  Argument 
for  finding  her  true  latitude. 

The  Sun's  mean  distance  from  the  ascending  node 
was  at  the  true  time  of  full  Moon,  Os.  4d.  49m.  35  sec- 
onds; but  it  appears  by  the  Example  that  the  true 
time  thereof,  was  6  hours,  33  minutes  and  38  seconds 
sooner,  than  the  mean  time,  and  therefore  we  must 
subtract  the  Sun's  motion  from  the  node  during  this 
interval,  from  the  above  mean  distance  Os.  4<1.  49m. and 
35  seconds,  in  order,  to  have  his  mean  distance  from 
the  node,  at  the  time  of  true  full  Moon.  Then,  to  this 
apply  the  equation  of  his  mean  distance  from  the  node, 
found  in  Table  13th,  by  his  mean  anomaly,  10s.  7d. 
27m.  45  seconds,  and  lastly,  add  six  signs,  and  the 
Moon's  true  distance  from  the  ascending  node,  will  be 
found  as  follows  ; — 


Stc.  17  Deliniation  of  Lunar  Eclipses.  239 

S      D      M      S 

Sun  from  node  at  mean  time  of  full  Moon,  0     4  49  35 

{6  hours,  15  35 
33  minutes,  1  26 
38  seconds. 2 

Subtract  the  sum,     -     -     -     -  "-     -     -     •_ 
Remains  his  mean  dist.  at  true  full  Moon,  0     4  32  32 

Equation  of  his  mean  distance,  add,  1  38     0 

Sun's  true  distance  from  the  node,  0     6  10  32 

To  which  add, 6000 

Moon's  true  distance  from  the  node,  6     6  10  32 

And  it  is  the  argument  used  to  find  her  true  latitude  at 
that  time.  Therefore,  with  this  Argument,  enter  Ta- 
ble 14th,  making  proportions  between  the  latitudes 
belonging  to  the  6th  and  7th  degree  of  the  Argument 
for  the  10  minutes,  and  32  seconds,  and  it  will  give  32 
minutes  and  21  seconds  for  the  Moon's  true  latitude, 
which  appears  by  the  Table  to  be  south  descending. 

To  find  the  angle  of  the  Moon's  visible  path  with  the 
ecliptic.  This  may  be  always  stated  at  5  degrees  and 
35  minutes  without  any  error  of  consequence,  in  the 
projection  of  either  Solar  or  Lunar  Eclipses. 

To  find  the  Moon's  true  horary  motion  from  the 
Sun.  With  their  respective  anomalies,  take  out  their 
horary  motions  from  Table  15th,  and  the  Sun's  horary 
motion,  subtracted  from  the  Moon's,  leaves  remaining 
the  Moon's  true  horary  motion  from  the  Sun,  in  the 
present  case,  30  minutes  and  52  seconds. 

The  above  elements  are  collected  for  use, 


240  Deliniation  of  Lunar  Eclipses.  Sec.  11 

D   H    M    s 
1st.  Tr ue  time  of  F.  Moon  in  May,  17628    3  50  50 

D 

2d.  Moon's  horizontal  parallax,                    0  57  23 

3d.  Sun's  semi-diameter,                                    15  56 

4th.  Moon's  semi-diameter,                               15  38 

5th.  S.  diameter  of  Earth's  shadow  at  Moon,  41  36 

6th.  Moon's  true  latitude  south  descending.     32  21 

7th.  Angle  of  Moon's  visible  path  with  eclp'tc  5  35  0 

8th.  Moon's  true  horary  motion  from  Sun,  0  30  52 

These  elements  being  found  for  the  construction  of 
the  Moon's  Eclipse  in  May,  1 762,  proceed  as  follows  : — 

Make  a  scale  of  any  convenient  length,  as  W.  X. 
and  divide  it  into  60  equal  parts,  each  part  standing  for 
a  minute  of  a  degree.  Draw  the  right  line  A.  C.  B, 
for  part  of  the  ecliptic,  and  C  A  perpendicular  thereto 
for  the  southern  part  of  its  axis,  (the  Moon  having 
south  latitude,) 

Add  the  semi-diameters  of  the  Moon  and  Earth's 
shadow  together,  which  in  this  case,  make  57  minutes 
and  14  seconds ;  and  take  this  from  the  scale  in  your 
compasses,  and  setting  one  foot  in  the  point  C.  as  a 
centre,  with  the  other  describe  the  semi-circle  S.  D.  B. 
in  one  point  of  which  the  Moon's  centre  will  be  at  the 
beginning  of  the  Eclipse,  and  the  other  at  the  end. 

Take  the  semi-diameter  of  the  Earth's  shadow,  (41 
minutes  and  36  seconds,)  in  your  compasses  from  the 
scale,  and  setting  one  foot  in  the  centre  C.  with  the 
other  describe  the  semi-circle  K.  L.  M.  for  the  south- 


Sec.  17  Deliniation  of  Lunar  Eclipses.  241 

ern  half  of  the  Earth's  shadow,  because  the  Moon's  lat- 
itude is  south  in  this  Eclipse. 

Make  C.  D.  equal  to  the  radius  of  aline  of  chords  en 
the  sector,  and  set  off  the  angle  of  the  Moon's  visible 
path  with  the  ecliptic,  (5  degrees  and  35  minutes,) 
fromD.  to  E.  and  draw  the  right  line  C.  F.  E.  for  the 
southern  half  of  the  axis  of  the  Moon's  orbit,  lying  to 
the  right  hand  from  the  axis  of  the  ecliptic  C.  A.  be- 
cause the  Moon's  latitude  is  south  descending  in  this 
Eclipse.  It  would  have  been  the  same  way  on  the 
other  side  of  the  ecliptic,  if  her  latitude  had  been  north 
descending,  but  contrary  in  both  cases,  if  her  latitude 
had  been  either  north,  or  south  ascending. 

Bisect  the  angle  A.  C.  E.  by  the  right  line  C.  g.  in 
in  which  the  true  equal  time  of  opposition  of  the  Sun 
aud  Moon  falls,  as  found  from  the  Tables. 

vTake  the  Moon's  latitude,  32  minutes  and  21  sec- 
onds, from  the  scale  in  your  compasses,  and  set  it  from 
C.  to  G.  in  the  line  C.  G.  g.  and  through  the  point  G. 
at  ridit  angles  to  C.  F.  E.  draw  the  rfeht  line  P.  H.  G. 

o  o  o 

F.  N.  for  the  path  of  the  Moon's  centre.  Then  F. 
shall  be  the  point  in  the  Earth's  shadow,  where  the 
Moon's  centre  is,  at  the  middle  of  the  Eclipse  ;  G.  the 
point  where  her  centre  is  at  the  tabular  time  of  her 
being  full ;  and  H.  the  point  where  her  centre  is,at  the 
instant  of  her  ecliptical  opposition. 

Take  the  Moon's  horary  motion  from  the  Sun,  (30 
minutes  and  52  seconds,)  in  your  compasses  from  the 
scale  \V.  X.  and  with  that  extent,  make  marks  along 
the  line  of  the  Moon's  path,  P.  G.  N.  then  divide  each 


242  Declination  of  Lunar  Eclipses,  Sec.  17 

space  from  mark  to  mark,  into  60  equal  parts,  or  hora- 
ry minutes,  and  set  the  hours  to  the  proper  dots  in  such 
manner,  that  the  dots  signifying  the  instant  of  full  Moon, 
namely,(50  minutes  and  50  seconds  after  3  in  the  mor- 
ning,) may  be  in  the  point  G.  where  the  line  of  the 
Moon's  path  enters  the  line  that  directs  the  angle 

D.  a  E. 

Take  the  Moon's  semi-diameter,  15  minutes  and  38 
seconds  in  your  compasses  from  the  scale,  and  with 
that  extent,  as  a  radius  upon  the  points  N.  F.  and  P.  as 
centres,  describe  the  circle  Q.  for  the  Moon  at  the  be- 
ginning of  the  Eclipse,  when  she  touches  the  Earth's 
shadow  at  V. ;  the  circle  R.  for  the  Moon  at  the  mid- 
dle, and  the  circle  S.  for  the  Moon  at  the  end  of  the 
Eclipse,  just  leaving  the  Earth's  shadow  at  W. 

The  point  N.  denotes  the  instant  when  the  Eclipse 
begins,  namely,  at  15  minutes  and  10  seconds  after  two 
in  the  morning.  The  point  F.  the  middle  of  the  Eclipse 
at  47  minutes  and  45  seconds  after  three,  and  the  point 
P.  the  3nd  of  the  Eclipse,  at  eighteen  minutes  after  five^ 
at  the  greatest  obscuration,  the  Moon  was  ten  digits 
eclipsed. 

The  Moon's  diameter,  (as  well  as  the  Sun's,)  is  sup- 
posed to  be  divided  into  12  equal  parts,  (called  digits,) 
and  so  many  of  these  parts  as  are  darkened  by  the 
Earth's  shadow,  so  many  digits  is  the  Moon  eclipsed. 
All  that  the  Moon  is  eclipsed  above  12  digits,  show  how 
far  the  shadow  of  the  Earth  is  over  the  body  cf  the 
Moon,  on  that  edge,  to  which  she  is  nearest,  at  the 
middle  of  the  Eclipse, 


Sec.  17  Declination  of  Lunar  Eclipses.  243 

It  is  difficult  to  observe  exactly,  either  the  beginning 
or  ending  of  a  Lunar  Eclipse,  even  with  a  good  Tele- 
scope ;  because  the  Earth's  shadow  is  so  faint,  and  ill- 
defined  about  the  edges,  that  when  the  Moon  is  either 
just  touching  or  leaving  it,  the  obscuration^of  her  limb 
is  scarcely  sensible,  and  therefore  the  closest  observers 
can  hardly  be  certain  to  four  or  five  seconds  of  time. 

But  both  the  beginning  and  ending  of  Solar  Eclipses, 
are  instantaneously  visible,  for  the  moment  that  the 
edge  of  the  Moon's  disk  touches  the  Sun's,  his  round- 
ness appears  to  be  broken  on  that  part,  and  the  moment 
she  leaves  it,  he  appears  perfectly  round  again. 

In  Astronomy,  Eclipses  of  the  MoorTare  of  great  use ' 
in  ascertaining  the  periods  of  her  motions,  especially 
such  Eclipses  as  are  observed  to  be  alike  in  all  circum- 
stances, and  have  long  intervals  of  time  between  them. 
In  Geography,  the  longitude  of  places  are  found  by 
Eclipses.  The  Eclipses  of  the  Moon  are  more  useful 
for  this  purpose,  than  those  of  the  Sun  ;  because  they 
are  more  frequeutly  visible,  and  the  same  Lunar  Eclipse 
is  equally  large,  at  all  places  where  it  is  seen. 

In  Chronology,  both  Solar  and  Lunar  Eclipses  serve 
to  determine  exactly  the  time  of  any  past  event,*  for 
there    are   so  many   particulars  observable   in   every 
Eclipse  with  respect  to  its  quantity — the  places  where  it 
is  perceivable,  (if  of  the  Sun,)  and  the  time  of  the  day, 
or  night,  that  it  is  impossible  that  there  can  be  two  So- 
lar Eclipses  in  the  course  of  many  ages,  which  are  alike 
in  all  circumstances.     From  the  preceding  explanation 
of  the  doctrine  of  Eclipses,  it  is  evident  that  the  dark- 


244  Ddiniation  of  Lunar  Eclipses.  Sec.  17 

ness  at  the  CRUCIFIXION  of  our  SAVIOUR  was  not  occa- 
sioned by  an  .Eclipse  of  the  Sun.  For  he  suffered  on 
the  day  on  which  the  PASSOVER  was  eaten  by  the  JEWS, 
namely,  the  thlrJ  day  of  April,  A.  D.  33  :  on  that  day» 
it  was  impossible  that  the  Moon's  shadow  could  fall  on 
the  Earth's.  For  the  JEWS  kept  the  PASSOVER  at  the 
time  of  full  Moon  ;  nor  does  the  darkness  in  total  Eclip- 
ses of  the  Sun,  last  above  four  minutes  and  six  seconds 
in  any  place  ;  whereas  the  darkness  at  the  CRUCIFIXION 
Jested  three  hours,  and  overspread,  at  least,  ail  the  Land 
of  JUDEA.  / 


Sec.  17 


Deliniatlon  of  Lunar  Eclipses. 


245 


SECTION  EIGHTEENTH. 


THE  FIXED 


THE  Stars  are  said  to  be  fixed,  because  'they  have 
been  generally  observed  to  keep  at  the  same  distances 
from  each  other  ;  their  apparent  diurnal  revolutions 
being  caused  solely  by  the  Earth's  turning  on  its  axis. 
They  appear  of  a  sensible  magnitude  to  the  eye,  because 
the  retina  is  affected,  not  onl}  by  the  rays  of  Jight  which 
are  remitted  directly  from  them,  but  by  many  thou- 
sands more,  which,  falling  upon  our  eye-lids,  and  upon 
the  aerial  particles  about  us,  are  reflected  into  our  eyes 
so  strongly,  as  to  excite  vibrations,  not  only  in  those 
points  of  the  retina,  where  the  real  images  of  the  stars 
are  formed,  but  also  in  other  points  of  some  distance 
round.  This  makes  us  imagine  the  Stars  to  be  much 
larger  than  they  would  appear,  if  we  saw  them  only 
by  the  few  rays  which  come  directly  from  them,  so  as 
to  enter  our  eyes,  without  being  intermixed  with  oth- 
ers. Any  person  may  be  sensible  of  this,  by  looking  at 
a  Star  of  the  first  magnitude,  through  a  long,  narrow 


24,8  Of  the  Fixed  Stars.  Sec.  1 8 

tube,  which,  though  it  takes  in  as  much  of  the  sky  as 
would  hold  a  thousand  such  stars,  yet  scarcely  renders 
that  one  visible. 

The  more  a  telescope  magnifies,  the  less  is  the  aper- 
ture through  which  the  star  is  seen ;  and  consequently 
the  less  number  of  rays  it  admits  into  the  eye.  The 
stars  appear  less  in  a  telescope  which  magnifies  200 
times,  than  they  do  to  the  naked  eye  ;  insomuch  that 
they  seem  to  be  only  indivisible  points ;  it  proves  at 
once  that  the  stars  are  at  immense  distances  from  us, 
and  that  they  shine  by  their  own  proper  light.  If  they 
shone  by  reflection,  they  would  be  as  invisible  -without 
telescopes,  as  the  satellites  of  Jupiter.  These  eatclliics 
appear  larger  when  viewed  with  a  go;>d  telescope,  than 
any  of  the  fixed  stars. 

The  number  of  stars  discoverable  in  cither  hr nil- 
sphere  by  the  unaided  sight,  is  not  above  a  thousand. 
This  at  first,  may  appear  incr<  ditable :  because  they 
seem  to  herd  most  innumerable,  but  the  deception  ari- 
ses from  our  looking  confusedly  upon  them  without  re- 
ducing them  to  any  order  :  look  steadfastly  upon  a  large 
portion  of  the  sky,  and  count  the  number  of  stars  in  if, 
and  you  will  be  surprised  to  find  thorn  so  few.  Consid- 
er only  how  seldom  the  Moon's  passes  between  us  and 
any  star,  (although  there  are  as  many  about  her  path, 
as  in  any  other  parts  of  the  Heavens  •,)  and  you  will 
soon  be  convinced  that  the  stars  are  much  thinner  sown,- 
than  you  expected.  The  British  catalogue,  which,  be- 
sides the  stars  visible  to  the  naked  eye,  includes  a  great 
number  which  cannot  be  seen,  without  the  assistance 


Sec.  18  Of  the  Fixed  Star*.  249 

of  a  telescope,  contains  no  more  than  three  thousand  in 
bath  hemispheres. 

As  we  have  incomparably  more  light  from  the  Moon, 
than  from  all  the  stars  together,  it  is  the  greatest  absurd- 
ity to  imagine,  that  the  stars  were  made  for  no  other 
purpose  than  to  cast  a  faint  light  upon  the  earth  ;  espe- 
cially, since  many  more  require  the  assistance  of  a  good 
telescope  to  find  them  out,  than  are  visible  without  that 
instrument.  Our  Sun  is  surrounded  by  a  system  of  plan- 
ets, and  comets,  all  of  which  would  be  invisible  from 
the  nearest  fixed  star  :  And,  from  what  we  already  know 
of  the  immense  distance  of  the  star?,  the  nearest  may  be 
computed  at  32  billions  of  miles  from  us,  which  is  far- 
ther than  a  cannon  ball  can  fly  in  7  millions  of  years, 
though  it  proceeded  \\  ith  the  same  velocity  as  at  its  first 
discharge.  Hence  it  is  easy  to  prove,  that  the  £un,  seen 
from  such  a  distance  would  appear  no  larger  than  a  star 
of  the  first  magnitude.  From  the  foregoing  observations 
ir  is  highly  probable,  thai  each  star  is  the  centre  of  a 
Bftgm&cenl  system  of  worlds,  moving  round  it,  though 
unseen  by  us,  and  are  irradiated  by  its  beams:  espe- 
cially, as  the  doctrine  of  plurality  of  worlds  is  rational, 
a *id  ^re.itly  manifests  the  power,  wisdom  and  goodness 
of  the  great  Creator. 

The  stirs,  on  account  of  their  apparently  various 
mignitudes  have  been  distributed  iutj  several  classes, 
or  orders.  Those  which  appear  largest  are  called  stars 
of  the  first  magnitude,  the  next  to  them  in  lustre,  stars 
of  the  second  magnitude,  and  so  on  to  the  sixth,  which 
are  the  smallest  that  are  visible  to  the  unaided  sight. — 

E* 


250  Of  the  Fixed  Stars.  Sec.  18 

This  distribution,  having  been  undo  long  before  the  in- 
vention of  telescopes,  die  stars  which  cannot  he  se*  n 
without  the  assistance  of  these  instruments,  are  distin- 
guished by  the  name  of  telescopic  stars. 

The'ancients  divided  the  starry  spheres  info  particu- 
lar constellations,  or  systems  of  stars,  5i<«T('ii  g  as  ih<  y 
Jay  near  each  other,  so  as  to  occupy  those  '  p  K  es  wl  u  h 
the  figures  of  different  sorts  of  animal.*,  or  things  would 
take  up,  if  they  were  there  delineated.  And  those  stars 
which  could  not  be  brought  into  any  particular  constel- 
lation, were  called  unformed  stars. 

This  division  into  different  cousteilationscr  asteri<ms. 
serves  to  distinguish  them  from  each  oih<  r;  so  tluit  any 
particular  star  may  be  readily. found  in  the  [leavens,  by 
menus  <»f  a  C(  Icstial  globe,  on  which  the  constellations 
arnso  delineated,  as  to  put  the  most  r<  markahie  stars 
into  such  parts  of  the  figures,  as  are  most  easily  dis- 
tinguished. The  number  of  ancient  constellations  is 
43,  and  upon  cur  present  globes,  about  70.  There  is 
also  a  division  of  the  Heavens  into  three  parts.  First 
the  Zodiac,  signifying  an  animal,  because  most  of  tl  e 
constellations  in  it,  which  are  twelve  in  number,  ase 
the  figures  of  animals,  as  Aries,  the  ram,  Taurus,  the 
bull,  Gemini,  the  twins,  Cancer,  the  crab,  Leo,  the  iirr. 
Virgo,  the  virgin,  Libra,  the  balance,  £corpc,  ihe 
scorpion,  Sagitaiius,  the  archer,  Capricornus,  the  gr/at, 

Aquarius,  the  water-bearer,  and  Pisce?,  the  fishes. 

The  Zudiac  goes  quite  rcnnd  the  Heavens,  it  is  ab:  ut 
16  degrees  broad,  so  that  it  takes  in  the  orbits  (  f  the 
MOOD,  and  .of  all  the  planets,  (excepting  that  cf  Pallas, 


Ses.  1 3  Of  Ihz  Fixzd  Stars.  2  5 1  * 

and  the  satellites  of  ilerschel.)  Along  ths-  nil. I. lie  of 
this  zone,  or  belt,  is  the  ecliptic,  or  circle  which  t!:e 
earth  cl  scribes  annually,  as  seen  from  the  Sun,  ai.d 
whk'h  the  ^un  appears  to  describe  as  seen  from  the 
.  earth.  Second.  All  that  region  of  the  Hea\  ens  which 
ii  on  tlic  north  side  of  the  Zodiac,  containing  21  eoi> 
ste-i.itions  :  And,  Third,  that  region  en  the  south  side 
of  the  ZoJiffc,  containing  15  constellations. 

There  is  a  remarkable  track  around  the  Heavens, 
called  the  Galaxy,  or  Milky  Way  from  its  peculiar 
whiteness.  It  was  formerly  thought  to  be  owing  to  a 
vast  number  of  very  small  stars,  closely  CGiinected,anil 
the  observation  of  Dr.  Her^chel  have  fully  confirmed 
the  opinion.  lie  therefore  considers  the  Galaxy  as  a 
very  extensive  brandling  congeries  of  many  millions 
of  stars,  which  prcbibly  owes  its  origin  to  several  re- 
markable large,  as  well  as  very  closely  scattered  small 
stars,  that  may  have  drawn  together  the  rest. 

OX  GROUPS  OF  STARS. 

Groups  of  Stars,  succeed  to  clustering  Stars  in  Dr. 
I  erschel's  arrangement.  A  group  is  a  collection  of 
Stars,  clcsely,  and  almost  equally  compressed,  and  of 
any  iigure  or  outline.  There  is  no  particular  conden- 
Kiii.ni  oft!).'  Star,;  to  in  lijate  tho  exis:e:i.;c  of  a  central 
f/rce,  ami  ll.e;  r  q-s  i^re  jiifikitnlly  fcparatcd  from 
neighboringptars,  to  show  that  they  form  peculiar  sys- 
terns  of  tkeirown. 


252  Of  the  Fixed  Stars.  Sec.  1 S 

ON  CLUSTERS  OF  STARS, 

Dr.  Herschel  regards  Clusters  of  Stars  as  the  most 
magnificent  objects  in  the  Heavens.  They  differ  from 
groups  in  their  beautiful  and  artificial  arrangement. — 
Their  form  is  generally  round,  and  their  condensation 
is  such  as  to  produce  a  mottled  lustre,  somewhat  re- 
sembling a  nucleus.  The  whole  appearance  of  a  clus- 
ter indicates  the  existence  of  a  central  force,  residing 
either  in  a  central  body,  or  in  the  centre  of  gravity  of 
the  whole  system. 


Sec.  18       Interrogations  for  Section  Eighteenth.        253 


Interrogations  for  Section  Eighteenth. 


What  is  a  fixed  Star  1 

Why  do  they  appear  of  sensible  magnitude   to  the 
eye? 

Do  the  Stars  appear  larger  when  viewed  through  a 
telescope,  than  viewed  with  the  eye  only  1 

What  does  it  prove  ? 

Which  appear  the  largest,  the  satellites  of  Jupiter, 
or  the  Stars,  when  viewed  with  a  telescope  ? 

About  how  many  Stars  in  a  clear  night  can  be  seen 
by  the  naked  eye  ? 

How  many  in  the  British  catalogue  ? 
Are  some  of  that  number  telescopic  1 

Would  the  planets  and  comets  of  the  Solar  System 
be  invisible  from  the  nearest  Star  ? 

At  how  many  miles  distant  may  we  with  propriety, 
suppose  the  nearest  fixed  Star  ? 


254       Interrogations  far  Section  Eighteenth.       Fee.  18 

How  long  would  a  cannon  ball  be  in  flying  that  dis- 
tance, supposing  it  should  continue  to  move  with  the 
same  velocity,  as  at  its  first  discharge  ? 

Of  what  size  would  the  Sun  probably  appear  from 
the  nearest  fixed  Star  ? 

Is  it  not  probable  that  every  the  centre  of  a 

? 


•   Qn~3§Eiat  accoun.t  have   they  beei.  ;uted  into 

classes-? 

Whatare  those  called  which  appear  largest  ? 
What  are  Constellations  1 

What  is  the   use  of  dividing  them  into  Constella- 
tions ? 

How  many  Constellations  on  the  celestial  globes  ? 

What  is  the  Zodiac  1 

How  many  Constellations  in  the  Zodiac  ? 

What  is  the  breadth  of  the  Zodiac  ? 

What  the  Galaxy,  or  Milky  \\  ay  ? 

\\  hat  is  a  Group  of  Stars  1 

What  are  Clusters  of  Stars  1 


SECTION  NINETEENTH. 


AC  "3" XT  O7  THE  GREGORIAN,  OR  NEW  STYLF,  TCGETZIEH 
WITH    SDM5    CHRONOLOGICAL    PilOELSMP,    FOR 
FINDING  THE  EPACT,  GOLDEN  NUMB£I?, 
DOMINICAL  LETTER,  &C. 


POPE  GREGORY  THIRTEENTH,  made  a  reformation 
of  the  calendar.  ri  he  Julian  calender,  (or  Old  Style,) 
had  before  that  time,  been  in  general  use  all  over 
Europe.  The  year,  according  to  the  Julian  calendar 
consists  of  365  days  and  6  hours,  which  6  hours  being 
4  part  of  a  day,  the  common  years  consisted  of  365 
days  :  and  every  fourth  year,  one  day  was  added  to 
the  month  of  February,  which  made  each  of  thope 
years  consist  of 336  days,  commonly  called  leap  years, 

This   computation  ear  the  truth,)  is  m«re 

than  the  Solar   year,   by  1  1    minutes    and   3    seconds, 
whi ;•!>,  in  13] 

ill  E  [uirn  fen  da 

tim^of  the  pener  .:ii  of  N^re,  liefd  in  (;.       •    p 

3.,.^  of  the  Christian  Era,  to  the  time  of  Pope  Gregory, 
who  therefore  caused  ten  days  to  be  taken  cut  of  the 


250  Of  the  Gregorian  Calendar.  Sec.  19 

month  of  October,  1582,  to  make  the  Equinox  full  on 
the  2 1st  of  March,  as  it  dU  at  the  time  of  that  council ; 
and  to  prevent  the  like  variation  for  the  future,  he  or- 
dered that  three  days  should  be  abated  in  every  four 
hundred  years,  by  reducing  the  leap  year  at  the  close 
of  each  century,  for  three  successive  centuries,  to 
common  ysars,  and  retaining  the  leap  year  at  the  close 
of  each  fourth  century  only.  This,  at  that  time,  was 
esteemed  as  exactly  conformable  to  the  true  solar 
year.  But  since  that  time,  the  true  solar  year  is  found 
to  consist  of  365  days,  5  hours,  43  minutes  arid  49  sec- 
onds, which  in  50  centuries  will  make  another  day's 
variation. 

Though  the  Gregorian  Calendar,  (or  New  Sts  !e,) 
ha:l  long  been  in^  use  throughout  the  greatest  part  of 
Europe,  it  did  not  take  place  in  GREAT  BRITAIN  and 
AMERICA,  till  the  first  of  January,  1752,  and  in  Sep- 
tember following,  the  1 1  days  were  adjusted  by  calling 
the  third  day  of  that  month,  the  fourteenth,  and  contin- 
uing the  rest  in  their  order. 

CHRONOLOGICAL  PROBLEMS. 

As  there  are  three  leap  years  to  be  abated  in  every 
four  centuries,  to  find  which  century  is  to  be  leap  year 
and  which  not. 

RULE. — Cat  off  two  cyphers  from  the  given  year, 
and  divide  the  remaining  figures  by  4,  if  nothing  re- 
main, the  year  will  be  leap  year. 

The  year  4  i^^r  there  being  a  remainder  of  3,  it  will 
not  be  leap  year.  But  the  year  *~p^  will. 


Sec.  19  Chronological  Problems.  257 

To  find  the  Dominical  or  Sunday  Shelter. 

RULE.  —  To  the  given  year,  add  its  fourth  part,  re- 
jecting  remainders,  divide  the  sum  by  7,  and  if  there 
be  no  remainder,  A  is  the  Sunday  letter  ;  but  if  any 
number  remains,  then  the  letter  standing  under  that 
number,  is  the  Dominical  letter,  and  the  day  of  the 
week  on  which  the  year  commences. 

A  leap  year  has  two  Dominical  letters,  the  first  of 
which  commences  the  year,  and  continues  to  the  24th 
of  February,  and  the  other  to  the  end  of  the  year. 
EXAMPLE. 

1234557      Required  the  Dominical  letters 

for  the  years  1832. 
^  £      £  4)1832 

£  £  Jt  *  |  &%  458 

"I  H  1|  1  •%  §  Days  in  a  week  7)2290 

j  -o  jC  «  -  So*     i 

c  327  —  1 


_ 

0123456      The  year  1832,  was  leap  year,  §• 

A  G  F  E  D  C  B         according  to  the  work,  the  re- 

mainder being  1,  the  first   Sunday  letter  was  A,  and 

G  was  the  second,the  year  also  commenced  on  Sunday. 

To  find  tlie  Golden  Number. 
RULE.  —  Add  1  to  the  given  year,  divide  the  sum  by 
19,  and  the  remainder  will  be  the  Golden  Number  :  if 
nothing  remain,  then  19  will  be  the  number  sought. 
Required  the  Golden  Number  for  the  year  1832, 
To  the  given  year  1832 

Add  _  1 

19)1833(96 
171 


123 
114 


Golden  Number,   9 


258  Chronological  Problems  &*e.  19 


Toitsid 

RULE.  —  Subtract  1  from  the  Golden  Number,  divide 
the  remainder  by  3,  if  1  remain,  add  10  to  the  dividend, 
the  sum  will  be  the  Epact  ;  if  nothing  remain,  the  divi- 
dend is  the  Epact. 

Required  the  Epact  for  the  year  1832.  The  Golden 
Number,  as  found  above,  is  9,  therefore,  subtract  l,and 
the  remainder  is  8,  divide  8  by  3,  and  the  quotient  is 
two,  and  2  remains,  multiply  this  remainder  by  10, 
and  the  product  is  20,  to  which  add  the  dividend,  and 
the  sum  is  29,  the  Epact  for  1832. 

To  find  the  year  of  the  Dionysian  Period. 

RULE.  —  Add  to  the  given  year  457,  divide  the  sum 
by  532,  and  the  remainder  will  be  the  number  required. 

Required  the  year  of  the  Dionysian  Period,  for  the 
year  1832.     To  the  given  year,  1832. 
Add  457 

532)2289(4 
2128 

161  =Dionysian  Period. 

To  find  tiie  Julian  Period. 

RULE.  —  Add  4713  to  the  given  year,  and  the  sum 
will  be  the  Julian  Period. 

Required  the  Julian  Period  for  the  year  of  the 
Christian  Era,  1832.  1832 

4713 
6545  year  of  the  Julian  period. 

*To  find  tiie  Cyelo  cf  tlio  STZSI,  Goldca  Kwmljer,  aad  indiction.for  any 
Current  Year* 

Rule.  —  To  the  current  year  add  4713,  divide  the 
sum  by  28,  19,  and  15,  respectively,  and  the  several 
remainders  will  be  the  numbers  required.  If  nothing 
remains,  the  divisors  are  the  required  numbers. 

*  A  Cycle  is  a  perpetual  round,  or  circulation  ci  the  mine  parts  ol  lime 
of  an}7  sort.  The  Cycle  of  the  Sun,  is  a  revolution  of  28  years,  in  which 
the  days  of  the  mouths  return  again  to  the  same  days  of  the 


Sec.  19  Chronological  Problems.  259 

Required  the  Cycle  of  the  Sun,  Golden  Number,  «§r 
Indiction  for  the  year  1832. 

1832 
4713 


56 


94 

84 


1832 

4713 

1832 
4713 

19)6545(344 
67 
84 
76 

Io)6545(433 
60 

~oT 
45 

80 
76 

95 

SO 

105 

84 

2i=ivycle  of  the,  Sun    2=Golden  Number  5=lndiction. 

T»  find  on  vrSiot  day  Easier  vrSll  Ir.appeu. 

It  was  ordered  by  the  Nicene  Council,  that  Easter 
Sunday,  should  be  kept  on  the  first  Sunday  after  the 
1st  full  Moon  which  happened  upon,or  after  the  twenty- 
first  day  of  March,  the  day  on  which  they  thought  the 
Vernal  Equinox  happened,  though  this  was  a  mistake, 
for  the  vernal  equinox  that  year  fell  on  the  20th  of 
March  ;  but  yet,  the  full  Moon  which  fell  on,  or  next 
after  the  twenty-first  of  March,  they  called  the 
Paschal  full  Moon ;  and  by  the  introduction  of  the 
Gregorian,  or  New  Style,  the  equinox  will  new  always 
happen  on  the  twentieth,  or  twenty-first  of  March  :  and 
if  the  full  Moon  happen  on  a-  Sunday,  Easter  Day  is  to 
be  the  next  Sunday  after.  Therefore,  find  the  time  of 
the  next  full  Moon  after  the  21st.  of  March,  and  the 
following  Sunday  is  Easter. 

of  the  week,  the  Sun's  place  to  the  same  signs  and  degrees  of 
the  ecliptic,  on  the  same  months  and  days,  so  as  not  to  differ  one  degree 
in  an  hundred  years,  and  the  leap  years  begin  the  same  course  over  again, 
with  respect  to  the  days  of  the  week,  on  which  the  days  oflho  months  fall. 
The  Cycle  of  the  Moon,  (commonly  called  the  Golden  Number,)  is  a 
revolution  of  19  years,  in  which  the  conjunctions,  oppositions,  and  other 
aspects  of  the  Moon,  are  within  an  hour  and  a  half,  of  being  the  same  as  they 
were,  on  the  same  days  of  the  months  19  years  before.  The  indiction  is  a 
revolution  of  15  years",  used  only  by  the  ROMANS,  for  indicating  the  times 
of  certain  payments  made  by  the  subjects  of  the  REPUBLIC  :  It  was  es- 
tablished by  CONSTANTINE,  A.  D.  312, 


260 


Of  the  Fixed  Stars. 


Sec.  19 


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Sec.  1 9          Tabular  View  of  the  Solar  System.         26 1 


TABULAR  VIEW  OF  THE  SOLAR  SYSTEM, 


Names 

Mean  diam 

Mean  distan- 

The cor- 

Mean ap- 

Mean apparent 

of  the 

eters  in 

ces  from  the 

rect  mean 

parent  di- 

diameter as  seen 

Planets. 

Miles. 

Sun  in  round 

distances 

ameters  as 

from  the  Sun. 

numbers  of 

that  of  the 

seen  from 

Miles. 

earth  being 

the  earth. 

100000 

M      S  3DS. 

The  Sun 

883,246 

32    1     30 

Mercury 

3,224 

37,000,000 

38,710 

10 

16  seconds. 

Venus 

7,687 

68,000,000 

72,333 

0  58      0 

30 

Earth 

7,912 

95,000,000 

100,000 

17,2 

Moon 

2,180 

95,000,000 

100,000 

31     8 

4,6 

Mars 

4,189 

144,000,000 

152,369 

0  27      0 

10 

Vesta 

238 

225,000,000 

237,300 

0    0    30 

Juno 

1,425 

252,000,000 

265,700 

030 

Ceres 

163 

263,000,000 

276,500 

1      > 

1,024 

6,4  5 

Pallas 

80 
20,99 

265,000,000 

279,100 

>5l 
6,55 

Jupiter 

89,170 

490,000,000 

520,279 

0  39  0 

37 

Saturn 

79,042 

900,000,000 

954,072 

18 

16 

Herschel 

35,112 

1,800,000,000  1,908,352 

3  54 

4 

TABULAR  VIEW  OF  THE  SOLAR  SYSTEM. 


If  AMES 
OF  THE 

Tropical  Revo- 
lutions. 

Sydereal  Revo- 
lutions. 

Place  of  Aphe- 
lion in  Jan.  1800 

Planets. 

Sun 

S          I)         M         S 

D           II        31         S 

D        II       M         S 

Mercury 

87    23     14    32 

87    23    15    34 

8     14    20    50 

Venus 

224    16     41     27 

224     16     49     10 

10      7     59      1 

Earth 

365      5    48    49 

365      6      9     12 

9      8     40    12 

Moon 

Mars 

686    22     18    27 

686    23    30    35 

5      2    24      4 

Vesta 

1155      4 

2      9    42    53 

Juno 

1588 

I  7    29    49    33 

Ceres 

1681 

I  4    25    57     15 

Pallas 

1703     16     48        J10      1      7      0 

Jupiter 

4330     14    39      2 

4332     14    27     10!  6     11      8    20 

Saturn 

10746     19     16     15 

10759       1     51     11 

8    29      4     11 

Herschel  130637      400 

i 

30737     18      0      0 

11     16    30    31 

CONTENTS. 

PAGK. 

Astronomy  in  General,  7 

Interrogations  for  Section  First,  15 

Description  of  the  Solar  System,  17 

of  Mercury,  19 

of  Venus,  20 

Transits  of  Mercury,  22 

of  Venus,  25 

Description  of  the  Earth,  25 

of  the  Moon,  26 

of  Mars,  28 

of  Vesta,  30 

of  Juno,  31 

of  Ceres,  3t 

of  Pallas,  S3 

of  Jupiter, 

of  Saturn,  37 
of  Herschel,  or  Uranus, 

of  Comets,  42 
Interrogations  for  Section  Second, 

On  Gravity,  53 

Interrogations  for  Section  Third,  57 

Phenomena  of  the  Heavens,  as  seen  from  different  parts  of  the  Earth,  59 

Interrogations  for  Section  Fourth,  67 
Physical  Causes  of  the  Motions  of  the  Planets, 

Interrogations  for  Section  Fifth,  [  76 

Of  Light,  78 

Of  Ai?,  81 
Interrogations  for  Section  Sixth, 

To  find  the  distances  of  the  Planets  from  the  Sun,  87 
Interrogations  for  Section  Seventh, 

Of  the  Equation  of  Time,  £> 
Of  the  precession  of  the  Equinoxes, 
Of  the  Moon's  Phases, 

Of  the  Phenomena  of  the  Harvest  Moon,  *y|> 
Interrogations  for  Section  Ninth, 

On  Tides,  }J» 

Interrogations  for  Section  Tenth,  *** 
Astronomical  Problems, 

On  Eclipses,  }™ 

Interrogations  for  Section  Twelfth,  *?J 

On  the  Construction  of  theJAstronomical  Tables,  y>y 

Interrogations  for  Section  Thirteenth,  *,*4 
Directions  for  the  Calculation  of  Eclipses, 

Astronomical  Tables,  *°* 

Examples  for  the  Calculation  of  Eclipses,  2yt> 

To  find  tho  Sun's  true  place,  **" 

To  find  tho  Sun's  true  distance  from  the  Moon  s  ascending  node,  ~-J£ 
To  project  an  Eclipse  of  the  Sun, 

Projection  of  Lunar  Eclipses,  _^ 

On  the  Fixed  Stars,  i; ' 

On  Groups  of  Stars,  ^ 

Clusters  of  Stars,                                                                          , .  «E« 

Interrogations  for  Section  Eighteenth,  ~ 

An  account  of  the  Gregorian,  or  New  Style,  gg 
Chronological  Problems. 


• 


Ostrander, 

planetarium  and 


O 


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1  \ 


calculator 


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08 

1832 


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